Can I get some assistance on this function in precalculus?Use the remainder theorem to find the remainder when f(x) is divided by x+5.  Then use the factor theorem to determine whether x+5 is a...

Can I get some assistance on this function in precalculus?

Use the remainder theorem to find the remainder when f(x) is divided by x+5.  Then use the factor theorem to determine whether x+5 is a factor of f(x). 

f(x)  = x^4 + 5x^3 - 7x^2 - 32x + 15

Asked on by deekoch

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embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

Given `x^4+5x^3-7x^2-32x+15` :

The remainder theorem states that the remainder when f(x) is divided by (x-c) is f(c).

Thus the remainder of `f(x)=x^4+5x^3-7x^2-32x+15` when divided by (x+5) is f(-5)=0.

The factor theorem states that if the remainder when f(x) is divided by (x-c) is zero, then (x-c) is a factor.

Thus (x+5) is a factor.

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hala718's profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

`f(x)= x^4 + 5x^3 - 7x^2 -32x + 15 `

`==gt f(x)= (x^4+5x^3)- (7x^2 +32x -15)`

Now we will factor x^3 from the first two terms and then factor the quadratic equation ( last 3 terms.).

`==gt f(x)= x^3(x+5) - (7x-3)(x+5).`

`` Now we will factor (x+5) .

`==gt f(x)= (x+5) (x^3-(7x-3)) `

`==gt f(x)= (x+5)(x^3 -7x+3)`

`` We notice that (x+5) is a factor of f(x).

Then the remainder when f(x) is divided by (x+5) is zero because (x+5) is a factor of f(x).

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