Two cubic expressions are defined by $\mathrm{f}(x) = x^3 + (a - 3)x + 2b$ and $\mathrm{g}(x) = 3x^3 + x^2 + 5ax + 4b$ where $a$ and $b$ are constants.

(i)
Given that $\mathrm{f}(x)$ and $\mathrm{g}(x)$ have a common factor $(x + 3)$ , show that $a = -4$ and find the value of $b$ .
(ii)
Using the values of $a$ and $b$ , factorise $\mathrm{f}(x)$ completely. Hence, show that $\mathrm{f}(x)$ and $\mathrm{g}(x)$ have two common factors.

Remainder factor theorems Simultaneous one linear 中文

It won't just hand you the answer — it asks where you're stuck.How the checker works

Worked solution

9(i)

1. Apply the Factor Theorem to $\mathrm{f}(x)$ to form the first equation.

Since $(x + 3)$ is a factor of $\mathrm{f}(x)$ , $\mathrm{f}(-3) = 0$ .
$(-3)^3 + (a - 3)(-3) + 2b = 0$
$-27 - 3a + 9 + 2b = 0$
$-3a + 2b = 18 \quad \text{--- (1)}$

2. Apply the Factor Theorem to $\mathrm{g}(x)$ to form the second equation.

Since $(x + 3)$ is a factor of $\mathrm{g}(x)$ , $\mathrm{g}(-3) = 0$ .
$3(-3)^3 + (-3)^2 + 5a(-3) + 4b = 0$
$3(-27) + 9 - 15a + 4b = 0$
$-81 + 9 - 15a + 4b = 0$
$-15a + 4b = 72 \quad \text{--- (2)}$

3. Solve the simultaneous equations to find $a$ and $b$.

Multiply (1) by 2:
$-6a + 4b = 36 \quad \text{--- (3)}$
Subtract (3) from (2):
$(-15a + 4b) - (-6a + 4b) = 72 - 36$
$-9a = 36$
$a = -4 \quad \text{(shown)}$
Substitute $a = -4$ into (1):
$-3(-4) + 2b = 18$
$12 + 2b = 18$
$2b = 6$
$b = 3$

Answer

$a = -4$ , $b = 3$

AI analysis — may contain mistakes.

Worked this one out on paper? Snap your steps and Gigle's AI math checker will circle the step where it went wrong.

Two cubic expressions are defined by f(x)=x3+(a−3)x+2b\mathrm{f}(x) = x^3 + (a - 3)x + 2bf(x)=x3+(a−3)x+2b and g(x)=3x3+x2+5ax+4b\mathrm{g}(x) = 3x^3 + x^2 + 5ax + 4bg(x)=3x3+x2+5ax+4b where aaa and bbb are constants.

Worked solution

9(i)

More questions on this topic

Two cubic expressions are defined by $\mathrm{f}(x) = x^3 + (a - 3)x + 2b$ and $\mathrm{g}(x) = 3x^3 + x^2 + 5ax + 4b$ where $a$ and $b$ are constants.