Studdy AI Logo
Math  /  Algebra

QuestionUse the given zeros to write the complete factored form of f(x)f(x). f(x)=2x2−23x+66;f(x)=2 x^{2}-23 x+66 ; zeros: 112\frac{11}{2} and 6 f(x)=f(x)= □\square (Type your answer in factored form. Use integers or fractions for any numbers

Studdy Solution

STEP 1

1. We are given the quadratic function f(x)=2x2−23x+66 f(x) = 2x^2 - 23x + 66 .
2. The zeros of the function are 112 \frac{11}{2} and 6 6 .
3. We need to express f(x) f(x) in its complete factored form.

STEP 2

1. Use the zeros to write the factors of the polynomial.
2. Express the polynomial in factored form.
3. Verify the factored form by expanding it to ensure it matches the original polynomial.

STEP 3

Since the zeros of the polynomial are given as 112 \frac{11}{2} and 6 6 , the factors of the polynomial can be written as:
(x−112)and(x−6) (x - \frac{11}{2}) \quad \text{and} \quad (x - 6)

STEP 4

To express the polynomial in factored form, we need to account for the leading coefficient of the quadratic term, which is 2 2 . Therefore, the complete factored form is:
f(x)=2(x−112)(x−6) f(x) = 2(x - \frac{11}{2})(x - 6)

STEP 5

Verify the factored form by expanding:
f(x)=2(x−112)(x−6) f(x) = 2(x - \frac{11}{2})(x - 6)
First, expand the factors:
(x−112)(x−6)=x2−6x−112x+112×6 (x - \frac{11}{2})(x - 6) = x^2 - 6x - \frac{11}{2}x + \frac{11}{2} \times 6
Combine like terms:
=x2−(6+112)x+33 = x^2 - \left(6 + \frac{11}{2}\right)x + 33
=x2−232x+33 = x^2 - \frac{23}{2}x + 33
Now, multiply by 2 2 :
f(x)=2(x2−232x+33) f(x) = 2(x^2 - \frac{23}{2}x + 33)
=2x2−23x+66 = 2x^2 - 23x + 66
The expanded form matches the original polynomial, confirming the factored form is correct.
The complete factored form of f(x) f(x) is:
f(x)=2(x−112)(x−6) f(x) = 2(x - \frac{11}{2})(x - 6)

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord