Math  /  Algebra

QuestionA cubic function g(x)g(x) with integral coefficients has the following properties: g(3)=0,g(34)=0,(x+2)g(3)=0, g\left(-\frac{3}{4}\right)=0,(x+2) is a factor of g(x),g(1)=84g(x), g(1)=-84. Determine g(x)g(x).

Studdy Solution

STEP 1

1. The function g(x) g(x) is a cubic polynomial with integer coefficients.
2. The roots of g(x) g(x) include x=3 x = 3 , x=34 x = -\frac{3}{4} , and x=2 x = -2 .
3. The value of the polynomial at x=1 x = 1 is g(1)=84 g(1) = -84 .

STEP 2

1. Use the roots to express g(x) g(x) in factored form.
2. Adjust the factored form to ensure integer coefficients.
3. Use the condition g(1)=84 g(1) = -84 to determine the leading coefficient.
4. Write the complete expression for g(x) g(x) .

STEP 3

Since g(x) g(x) has roots x=3 x = 3 , x=34 x = -\frac{3}{4} , and x=2 x = -2 , we can express g(x) g(x) as:
g(x)=k(x3)(x+34)(x+2) g(x) = k(x - 3)\left(x + \frac{3}{4}\right)(x + 2)
where k k is a constant.

STEP 4

To ensure integer coefficients, multiply the factor (x+34) \left(x + \frac{3}{4}\right) by 4:
g(x)=k(x3)(4x+3)(x+2) g(x) = k(x - 3)(4x + 3)(x + 2)

STEP 5

Expand the expression:
First, expand (4x+3)(x+2) (4x + 3)(x + 2) :
(4x+3)(x+2)=4x2+8x+3x+6=4x2+11x+6 (4x + 3)(x + 2) = 4x^2 + 8x + 3x + 6 = 4x^2 + 11x + 6
Now, expand (x3)(4x2+11x+6) (x - 3)(4x^2 + 11x + 6) :
(x3)(4x2+11x+6)=x(4x2+11x+6)3(4x2+11x+6) (x - 3)(4x^2 + 11x + 6) = x(4x^2 + 11x + 6) - 3(4x^2 + 11x + 6)
=4x3+11x2+6x12x233x18 = 4x^3 + 11x^2 + 6x - 12x^2 - 33x - 18
=4x3x227x18 = 4x^3 - x^2 - 27x - 18
Thus, g(x)=k(4x3x227x18) g(x) = k(4x^3 - x^2 - 27x - 18) .

STEP 6

Use the condition g(1)=84 g(1) = -84 to find k k :
g(1)=k(4(1)3(1)227(1)18)=84 g(1) = k(4(1)^3 - (1)^2 - 27(1) - 18) = -84
g(1)=k(412718)=84 g(1) = k(4 - 1 - 27 - 18) = -84
g(1)=k(42)=84 g(1) = k(-42) = -84
k=8442=2 k = \frac{-84}{-42} = 2

STEP 7

Substitute k=2 k = 2 back into the expression for g(x) g(x) :
g(x)=2(4x3x227x18) g(x) = 2(4x^3 - x^2 - 27x - 18)
g(x)=8x32x254x36 g(x) = 8x^3 - 2x^2 - 54x - 36
The complete expression for g(x) g(x) is:
8x32x254x36 \boxed{8x^3 - 2x^2 - 54x - 36}

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