Factors and Zeros

Factors and Zeros

They are also the values of x that, when plugged into a function, cause the value of the function to be zero.

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Zeros are just x-intercepts

Multiple Choice

The graph of f(x) is shown. Which statement about f(x) is true?

• The zero of f(x) is -3

• The zero of f(x) is 2

• The zeros of f(x) are -1 and 1

• The zeros of f(x) are -6 and 6

The Zeros and Factors of a Function are related

​f(x)= (x-1) x² (x+3)³

Multiple Select

A polynomial function contains the factors x, x-3, and x+1. Which graphs could represent the polynomial function. SELECT ALL THAT APPLY

As we discussed earlier in the year, you can divide polynomials (using the box)

Just like in normal division, there could be a reminder, but there may not be one

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The remainder theorem

​When you divide a polynomial by another polynomial, there are few things to understand

• If there is no remainder, that means the polynomial you divided by is a FACTOR of the first polynomial

• If there is any remainder at all, then it's not a factor.

• Rather than doing the entire division, you could simply plug in the number associated with the factor. This would give you the remainder

For Example...

So then is (x+2) a factor?

The triangle

A question might be:

If (x-2) is a factor of the function f(x)= x³ + 7x² - 10x - 16. What must be true?

​A.) f(-2)=0

B.) f(0)=2

C.) f(2)=-2

D.) f(0)=-2

E.) f(2)=0

Multiple Choice

The function $$p\left(x\right)$$ has an x-intercept at 3. Which of the following, if $$p\left(x\right)$$ were divided by it, would have no remainder?

• x-2

• x-3

• x+2

• x+3

Multiple Choice

If the function $$p\left(x\right)$$ has a zero at 5, which of the following is a true statement?

• (x-5) is a factor of p(x)

• (x+5) is a factor of p(x)

• p(x) has a

  y-intercept at (0,5)

• p(x) has 3 real solutions

Multiple Choice

When $$g\left(x\right)=3x^3-7x^2+12x+20$$ is divided by x-4, the remainder is 148. Which of the following is true?

• g(4)=148

• g(148)=-4

• g(0)=4

• g(-4)=-148

Multiple Choice

Given the graph of g(x), which statement is true?

• $$g\left(x\right)$$ has no real solutions

• $$\frac{g\left(x\right)}{x+5}$$ has no remainder

• $$\frac{g\left(x\right)}{x+1}$$ has no remainder

• $$\frac{g\left(x\right)}{x-1}$$ has no remainder

Multiple Choice

For $$h\left(x\right)$$ , when x=5, $$h\left(5\right)=0$$ . Which of the following is a true statement, given this information

• (x-5) is a factor of h(x)

• (x+5) is a factor of h(x)

• h(x) has a zero at (0,5)

• h(x) has a y-intercept at -5