Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Given that
f(x)=bx^(3)+5x^(2)+2x-8 and
g(x)=bx^(3)+6x+4 has a common factor of
x-a, where
a is an integer, find the value of
b.

$2$ . Given that $\mathrm{f}(x)=b x^{3}+5 x^{2}+2 x-8$ and $\mathrm{g}(x)=b x^{3}+6 x+4$ has a common factor of $x-a$ , where $a$ is an integer, find the value of $b$ .

View step-by-step help

View step-by-step help

Factor using a quadratic pattern

Full solution

Q.

2

. Given that

\mathrm{f}(x)=b x^{3}+5 x^{2}+2 x-8

and

\mathrm{g}(x)=b x^{3}+6 x+4

has a common factor of

x-a

, where

a

is an integer, find the value of

b

.

Plug in $x=a$ : Since $f(x)$ and $g(x)$ have a common factor of $x-a$ , let's plug in $x=a$ into both functions and set them equal to zero. $\newline$ $f(a) = ba^3 + 5a^2 + 2a - 8 = 0$ $\newline$ $g(a) = ba^3 + 6a + 4 = 0$
Subtract to eliminate $b$ : Subtract the second equation from the first to eliminate $b$ . $(ba^3 + 5a^2 + 2a - 8) - (ba^3 + 6a + 4) = 0 - 0$ $5a^2 + 2a - 8 - 6a - 4 = 0$
Simplify the equation: Simplify the equation. $5a^2 - 4a - 12 = 0$
Factor the quadratic: Factor the quadratic equation. $\newline$ $(5a + 6)(a - 2) = 0$
Set equal to zero: Set each factor equal to zero and solve for $a$ . $5a + 6 = 0$ or $a - 2 = 0$ $a = -\frac{6}{5}$ or $a = 2$ Since $a$ is an integer, $a = 2$ .
Plug $a=2$ back: Plug $a = 2$ back into the equations for $f(x)$ and $g(x)$ to solve for $b$ . $\newline$ $f(2) = 2b(2)^3 + 5(2)^2 + 2(2) - 8 = 0$ $\newline$ $g(2) = 2b(2)^3 + 6(2) + 4 = 0$
Simplify both equations: Simplify both equations. $\newline$ $f(2) = 16b + 20 + 4 - 8 = 0$ $\newline$ $g(2) = 16b + 12 + 4 = 0$
Solve for b: Solve for b using the equation from g( $2$ ). $\newline$ $16b + 16 = 0$ $\newline$ $16b = -16$ $\newline$ $b = \frac{-16}{16}$ $\newline$ $b = -1$

More problems from Factor using a quadratic pattern

Question

Find the greatest common factor.

\newline

9v^3, 6v^3

\newline

\newline

Write your answer as a constant times a product of single variables raised to exponents.

\newline

\underline{\hspace{1cm}}

Posted 1 month ago

Question

Factor out the greatest common factor. If the greatest common factor is

1

, just retype the polynomial.

\newline

6x^3 + 9x^2

\newline

Posted 2 months ago

Question

\newline

k^2 + 9k + 18

\newline

Posted 1 month ago

Question

\newline

9j^2 - 16

\newline

Posted 1 month ago

Question

\newline

j^3 + 2j^2 + 5j + 10

Posted 2 months ago

Question

x^4 + 6x^2 + 9

\newline

All factors in your answer should have integer coefficients.

\newline

Posted 2 months ago

Question

\newline

z^2 + 13z + 22

\newline

Posted 1 month ago

Question

Simplify the expression:

\newline

5x + 9x - 4x - 2x

\newline

Posted 3 months ago

Question

3 - 3^2

Posted 3 months ago

Question

5 - 5^2

Posted 1 month ago