FINDING X AND Y - INTERCEPTS OF THE POLYNOMIAL FUNCTIONS || GRADE 10 MATHEMATICS Q2

Summary

本视频课程讲解了如何找到多项式函数的x轴和y轴截距。首先解释了y轴截距是函数输入值为零时的点，而x轴截距是函数输出值为零的点。然后通过几个例子演示了如何使用有理根定理找到截距，包括如何将多项式方程设为零并求解。视频通过具体例子详细展示了如何找到三次、四次等不同次数多项式的截距，并解释了这些截距如何影响图形的绘制。

Takeaways

• 📌 视频课程主要讨论了多项式函数的x轴截距和y轴截距的寻找方法。
• 🔍 y轴截距是函数在输入值为零时的点，即x=0时的函数值。
• 📐 x轴截距是函数输出值为零的点，多项式的最高次数为n，则x轴截距最多有n个。
• 📘 举例说明了如何找到多项式函数的截距，如函数y=x^3-6x^2+3x+10。
• 🔢 使用有理根定理来找到x轴截距，通过测试可能的有理数根。
• 📊 通过将x设置为0来找到y轴截距，即计算f(0)的值。
• 📈 展示了如何通过因式分解来简化寻找x轴截距的过程。
• 📑 提供了多个多项式函数的例子，包括不同次数的多项式。
• 🎓 强调了理解多项式函数的图形特性，如通过截距点来了解图形走向。
• 📝 视频最后鼓励观众学习使用有理根定理，并提供了练习题来巩固学习。

Q & A

什么是多项式函数的y截距？

多项式函数的y截距是函数在输入值为零时的点，即当x=0时，函数的输出值为y截距。

多项式函数的x截距是什么？

多项式函数的x截距是函数输出值为零时的点，即y=0时，解方程以找到x截距。

如何确定一个n次多项式的x截距的数量？

一个n次多项式最多有n个x截距，这意味着函数的图像最多与x轴交n次。

给定多项式y=x³-6x²+3x+10，如何找到其x截距？

可以使用有理根定理，通过代入可能的有理数根（如±1, ±2, ±5, ±10）来找到x截距。最后发现x= -1, 2和5是该多项式的x截距。

y=x³-6x²+3x+10的y截距是多少？

将x设为0，y截距为10，因此图像将通过(0, 10)这个点。

如何用分解方法求解多项式方程的x截距？

可以将多项式分解成线性因式，然后将每个因式等于零，解得x的值。例如y=x³-4x²+x+6可分解为(x+1)(x-2)(x-3)，所以x截距为-1, 2, 3。

什么是有理根定理？

有理根定理用于寻找多项式方程可能的有理数根，它将常数项的因数与最高次项的系数的因数进行组合，得到可能的有理根。

如何确定多项式的y截距？

要确定y截距，只需将x值设为零，代入多项式方程，计算得到的y值即为y截距。

什么是多项式的“转折点”？

转折点是函数图像由上升转为下降或由下降转为上升的点。对于一个n次多项式，最多有n-1个转折点。

多项式y=-x⁴+16的x截距和y截距分别是多少？

通过因式分解找到x截距为-2和2，y截距为16，因此图像将通过(0, 16)这个点。

Outlines

00:00

🔢 如何找到多项式函数的X和Y截距

本段介绍了如何找到多项式函数的X和Y截距。Y截距是在输入值为0时的函数值，而X截距则是输出值为0时的点。函数的X截距数量最多为多项式的次数。举例来说，三次多项式的X截距可能最多有三个。接着，提供了一个多项式的具体例子：Y = x³ - 6x² + 3x + 10，演示了如何利用有理根定理来求X截距与Y截距。

05:03

📉 继续计算多项式的X截距

本段进一步计算X截距，说明通过带入正负不同的值（如1、-1、2、-2、5、-5等）来验证结果。例如，当x = -1时，输出值为0，因此-1是一个X截距；类似地，x = 2和x = 5也是X截距。这意味着函数的图像将通过(-1, 0)、(2, 0)和(5, 0)这些点。Y截距则通过设x = 0来求解，得到Y截距为10，对应的点为(0, 10)。

10:29

🧮 使用有理根定理求X和Y截距

本段提供了另一个例子，Y = x³ - 4x² + x + 6，通过有理根定理和因式分解来求解X截距。方程因式分解为(x + 1)(x - 2)(x - 3)，因此X截距为-1、2和3。接着，设x = 0来求解Y截距，得到Y截距为6。图像将通过这些X截距和Y截距点：(-1, 0)、(2, 0)、(3, 0)和(0, 6)。

15:30

📝 使用特殊因式分解法求解X截距

本段讨论了如何通过因式分解多项式Y = -x⁴ + 16来求解截距。利用平方差公式将多项式因式分解为(x² + 4)(x² - 4)，再将后者进一步因式分解为(x + 2)(x - 2)。X截距为-2和2，而Y截距通过设x = 0得出Y = 16，对应的点为(0, 16)。

📊 分析多项式的X和Y截距

本段最后一个例子展示了多项式Y = 2x⁴ + 8x³ + 4x² - 8x - 16的截距计算。通过因式分解，得出X截距为-1、1和-3。最后通过设x = 0来求解Y截距，得到Y = -6，对应的点为(0, -6)。

Keywords

💡多项式函数

多项式函数是由变量和系数组成的一种数学表达式，形式为a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0。视频中讨论了如何找到多项式函数的x和y截距。

💡y截距

y截距是函数图像与y轴相交的点，即当x=0时函数的值。视频中解释了如何通过将x设为0来找到多项式函数的y截距。

💡x截距

x截距是函数图像与x轴相交的点，即函数值为0时的x值。视频中说明了多项式函数的x截距可以通过求解方程f(x)=0来找到。

💡多项式的次数

多项式的次数是指最高次幂的指数。视频中提到次数为n的多项式最多有n个x截距和n-1个转折点。

💡理性根定理

理性根定理用于确定多项式方程的有理根。视频中使用理性根定理来找出多项式函数的可能x截距。

💡因式分解

因式分解是将多项式分解为几个多项式的乘积。视频中多次使用因式分解方法来简化多项式并求解x截距。

💡常数项

常数项是多项式中没有变量的项。在视频中，常数项用于理性根定理的应用，帮助找到多项式的有理根。

💡系数

系数是多项式中变量前的数字。视频中讨论了如何通过系数来确定可能的x截距。

💡转折点

转折点是多项式函数图像的局部极值点。视频中提到一个n次多项式最多有n-1个转折点。

💡平方根法

平方根法是解二次方程的一种方法，通过将方程转化为完全平方形式并求解。视频中提到在因式分解时使用平方根法来解决二次方程。

Highlights

Introduction to finding x and y intercepts of polynomial functions.

The y-intercept occurs where the input value is zero (x = 0).

The x-intercepts occur where the output value is zero.

For a polynomial of degree n, the number of x-intercepts is at most n.

Example 1: Finding intercepts of the polynomial y = x³ - 6x² + 3x + 10.

Using the rational root theorem to find possible values for x-intercepts.

The x-intercepts for the polynomial y = x³ - 6x² + 3x + 10 are found to be -1, 2, and 5.

The y-intercept for this polynomial is 10, meaning the graph passes through (0, 10).

Example 2: Finding intercepts for the polynomial y = x³ - 4x² + x + 6.

Factoring the polynomial to find x-intercepts as -1, 2, and 3.

The y-intercept for this polynomial is 6, meaning the graph passes through (0, 6).

Example 3: Finding intercepts for the polynomial y = -x⁴ + 16.

Using difference of squares to factor the polynomial and find the x-intercepts at -2 and 2.

The y-intercept for this polynomial is 16, meaning the graph passes through (0, 16).

Example 4: Finding intercepts for the polynomial y = 2x⁴ + 8x³ + 4x² - 8x - 16 using factorization.

The x-intercepts are found to be -3, -1, and 1, and the y-intercept is -6.

Transcripts

[Music]

good day everyone

in this video lesson we will discuss

about

finding x and y intercepts of the

polynomial functions

so first the y-intercept of a polynomial

function is the point where the function

has an

input value of zero so kappa goku had

time

y intercept

nothing x as equal to zero so ib sub

n y intercept

not n the x intercepts are the points

where the

output value is zero a polynomial of

degree n

will have at most n so e big sub n

let's say the polynomial function in

degree is three

so possible namakua nothing uh

x-intercept

is at most three then putting three

padding two

or padding is on x-intercept line so

depending

okay and then uh n minus one

is the turning point so it is

x and y intercept given

the polynomial functions

okay first find the intercepts of

y is equal to x cubed minus six

x squared plus three x plus ten

so these uh example number one

so yeah play nothing in the first

quarter

you're using the rationale

uh

1 negative 1 2

negative 2 5 negative 5

and 10 negative 10 and then kunin did

not

know uh factors now

leading coefficient not n so your

leading coefficient not in detail

so since it doing leading term not then

is necessary standard form so since it

young leading term not and so in leading

coefficient nothing is one

so an impossible factor is no one in

positive

and negative factors later that is 1 and

negative 1.

so after makuhana 10 you factor some

constant term

and then your leading coefficient

[Music]

like for example this one one divide one

so

the hat moon i did divide k1 so

for example one divide one so that is

one

negative one divided one that is

negative one two

uh divide one that is two staples k

negative number one one divide negative

one that is negative one since melon

bases you know i represent you so

and gargoyle n so after margarine

function at n okay so pakistan

and then papalito nothing in y is

non-zero

okay nothing in white and zero

so again it's a subtitle

function so simulant let's say start

target a positive one

so pakistan is a positive one little

okay so one cubed minus six times one

squared plus three times one plus ten

okay

an x-intercept so proceed take a

negative one

so uh synaptic net is a negative one so

negative one

cubed minus six times negative one

squared plus three times negative one

plus ten

that is equal to zero so since neg

equals

zero to negative one so e big sub hand

see negative one

i is a x intercept by the way

elanian possible in the x-intercept

indeed is a given sincere degree nothing

is three

so at most three among one at ten okay

padding

that long x intercept but in the lower

or padding issa

okay so since marinette is sung negative

one

so i'm pretty

okay so next time positive positive two

so pakistan's a positive two that is two

cubed minus six times two squared plus

three times two

plus ten the answer is zero

so eb sub ends the positive two ion two

i

casamasa x intercept so l again nothing

unboxed

next i opposite dial so okay negative

two

so copper is enough to choose that is a

negative two so independence is a hand

uh you can check using your calculator

or you can compute manually

equals to 20 negative 28 so a big sub

ends a negative to a hindi casama

proceed to okay pipe

pipe cube minus 6 times 5 squared plus 3

times pi

plus 10 the answer is zero so eb

submission positive

in class

zero impossible the x intercept

say negative 5 10 seconds negative 10

capacitors

rational

the x-intercepts are so negative 1

positive 2 and positive 5 so this means

that the graph will pass through

so negative 1 0 2 0

and 5 0 okay

and then x intercept

number y-intercept so not including

y-intercept

and going set x is equal to zero so

e-bikes have been populated

on variable x naught and non-zero so

y's that is our polynomial function so

papadi tending at n and x nothing and

zero

so y is equal to zero cubed minus six

times zero

so i'm sure cathy and class next

substitute

so therefore y is equal to 10

and that is the y-intercept is 10 this

means that the graph will also pass

through

as a

0 5 0 and 0 10

okay so i hope 19 behind using the

rational root

theorem okay positive example number two

find the intercepts of y is equal to x

cubed minus

four x squared plus x plus six so

determine gamma two

uh factored form so sonic ordinance

different uh types known special

products at young method company

factor no polynomials

um example so

given a polynomial function at n and

that

is y is equal to x cubed

minus four x odd and that is x

plus one so ethernet factored form

newton polynomial function at n

so you know are using the

factoring so patting yoga meetings

x cubed minus four x squared plus x plus

six

so to find the x intercept papadi

tandoor nothing and y is equal to zero

and to solve for

x so equate lag

zero so first x plus one is equal to

zero

x is equal to negative one bucket so the

path like nothing c one is a right side

so since positive magma bargain and sink

and negative one

x minus two is equal to zero so the

answer is x is

equal to two x minus three is equal to

zero the answer is

x is equal to three okay so

it only only x intercepts na tensa

polynomial function at also the

x-intercepts

are negative 1 2 and 3. so

this means that the graph will pass

through negative 1

0 2 0 and 3 0

and to find the y-intercepts set x is

equal to zero so papadi

is variable x

and the answer is six

so it picks up the end the y-intercept

is 6

this means that the graph will also pass

through

0 6.

another example we have find the

intercepts of y

is equal to negative x to the fourth

plus 16.

okay so using factored form so an in

factored formula

so d padding nothing i top edit on i

don't know difference of

uh two square okay you apply nathaniel

difference of two squares so

since many times negative so uh

nothing factoring nothing by negative

one so but i'm again x to the fourth

minus sixteen okay

so that is x squared plus 4

and x squared minus 4. okay

so after this

so you see x squared plus 4 in the

international

so the square root of 4 is 2 x plus 2

times x minus 2. so it only union

factored form

negative x to the fourth plus 16.

nothing so divide both sides by negative

one since my negative tile

so the answer is x squared plus four so

apply nothing you know

solving quadratic equation

you need to solve dial this is a

by using an tower nothing done

by using the square root method

pedinating apply on so it on formality

right side so maggie negative 4

in squared on both sides negative for

inside the radical sign imaginary

so eb sabine x plus four no

x squared plus four i hinting

value unit imaginary

zero so our x is equal to negative two

so again dito class so example nato

young degree num polynomials not n is

two okay so that is

so positive negative two and positive

too long i'm pretty not

in kuninjan so the x intercepts are

negative two

and two this means that the graph will

pass through negative two zero

and two zero so in the y-intercept

nothing is zero x so y is equal to

sixteen or the y-intercept is sixteen

and that is the gra also the graph will

also pass through 0

16. last example

find the intercepts of y is equal to 2 x

to the fourth

plus 8 x cubed plus four x squared minus

eight x

minus sixteen so um

example class

factored form or predicate class

uh

forms a big name factored formula and

that is two

times x plus one times x plus one

times x minus one times x plus three so

since your uh degree non-polynomial is

nothing is four so possible

x intercepts

are equal to zero and then equate

your factored formula tends as zero so x

plus one

is equal to negative one since same

number

i assumed so young excellent is negative

one

and then x minus one is equal to zero

x is equal to positive one x plus three

is equal to zero

x is equal to negative three so

therefore

x intercepts

no uh multiplicity national uh

announcement so therefore the x

intercepts are

negative 1 1 and negative 3.

this means that the graph will pass

through negative 1 0

1 0 and negative three zero so going

intense to find y

intercept a polytechnic x variable

nothing and zero

so y is equal to negative six

or the y intercept is negative six this

means that the graph will also pass

through

zero and negative six

thank you for watching this video i hope

you learned something

don't forget to like subscribe and hit

the bell button

put updated ko for more video tutorial

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