Scottish Highers Maths Polynomials: 10. (a) Show that $(x + 5)$ is a

10. (a) Show that $(x + 5)$ is a factor of $x^4 + 3x^3 - 7x^2 + 9x - 30$ - Scottish Highers Maths - Question 10 - 2023

Question 10

10. (a) Show that $(x + 5)$ is a factor of $x^4 + 3x^3 - 7x^2 + 9x - 30$. (b) Hence, or otherwise, solve $x^4 + 3x^3 - 7x^2 + 9x - 30 = 0$, $x \in \mathbb{R}$.

Worked Solution & Example Answer:10. (a) Show that $(x + 5)$ is a factor of $x^4 + 3x^3 - 7x^2 + 9x - 30$ - Scottish Highers Maths - Question 10 - 2023

Step 1

Show that $(x + 5)$ is a factor of $x^4 + 3x^3 - 7x^2 + 9x - 30$

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Answer

To show that $(x + 5)$ is a factor of the polynomial, we will use synthetic division.

Substitute $x = -5$ into the polynomial:

$f(-5) = (-5)^4 + 3(-5)^3 - 7(-5)^2 + 9(-5) - 30$ $= 625 - 375 - 175 - 45 - 30$ $= 625 - 375 = 250$ $250 - 175 = 75$ $75 - 45 = 30$ $30 - 30 = 0$
Since $f(-5)$ equals zero, by the Factor Theorem, $(x + 5)$ is indeed a factor of the polynomial $x^4 + 3x^3 - 7x^2 + 9x - 30$ .

Step 2

Hence, or otherwise, solve $x^4 + 3x^3 - 7x^2 + 9x - 30 = 0$, $x \in \mathbb{R}$.

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Answer

Given that $(x + 5)$ is a factor, we can perform polynomial long division to find the other factor:

Dividing $x^4 + 3x^3 - 7x^2 + 9x - 30$ by $(x + 5)$ yields:

Resulting Polynomial: $x^3 - 2x^2 + 3x - 6$
To solve for $x$ , we can set the resulting polynomial equal to zero:

$x^3 - 2x^2 + 3x - 6 = 0$
We can attempt to factor the cubic polynomial. Testing possible rational roots, we find:
- Testing $x = 2$ : $2^3 - 2(2^2) + 3(2) - 6 = 8 - 8 + 6 - 6 = 0$
- Thus, $x - 2$ is also a factor.
- Remaining factor after division is $(x^2 + 3)$ .
Now, solving the equations: $x - 2 = 0 \Rightarrow x = 2$ $x^2 + 3 = 0 \Rightarrow x^2 = -3 \Rightarrow x = \pm i\sqrt{3}$

Thus, the real solution is $x = 2$ .

10. (a) Show that $(x + 5)$ is a factor of $x^4 + 3x^3 - 7x^2 + 9x - 30$ - Scottish Highers Maths - Question 10 - 2023

Worked Solution & Example Answer:10. (a) Show that $(x + 5)$ is a factor of $x^4 + 3x^3 - 7x^2 + 9x - 30$ - Scottish Highers Maths - Question 10 - 2023

Show that $(x + 5)$ is a factor of $x^4 + 3x^3 - 7x^2 + 9x - 30$

Hence, or otherwise, solve $x^4 + 3x^3 - 7x^2 + 9x - 30 = 0$, $x \in \mathbb{R}$.

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