Scottish Highers Maths Polynomials: (a) Show that $(x

(a) Show that $(x - 1)$ is a factor of $f(x) = 2x^3 - 5x^2 + x + 2$ - Scottish Highers Maths - Question 2 - 2017

Question 2

(a) Show that $(x - 1)$ is a factor of $f(x) = 2x^3 - 5x^2 + x + 2$. (b) Hence, or otherwise, solve $f(x) = 0$.

Worked Solution & Example Answer:(a) Show that $(x - 1)$ is a factor of $f(x) = 2x^3 - 5x^2 + x + 2$ - Scottish Highers Maths - Question 2 - 2017

Step 1

(a) Show that $(x - 1)$ is a factor of $f(x) = 2x^3 - 5x^2 + x + 2$.

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Answer

To determine whether $(x - 1)$ is a factor of the polynomial $f(x)$ , we can use the Factor Theorem. According to this theorem, $(x - c)$ is a factor of $f(x)$ if and only if $f(c) = 0$ .

We will substitute $x = 1$ into the polynomial:

$f(1) = 2(1)^3 - 5(1)^2 + (1) + 2$
$= 2 - 5 + 1 + 2$
$= 0$

Since $f(1) = 0$ , it follows that $(x - 1)$ is indeed a factor of $f(x)$ .

Step 2

(b) Hence, or otherwise, solve $f(x) = 0$.

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Answer

Now that we know $(x - 1)$ is a factor, we can perform polynomial long division of $f(x)$ by $(x - 1)$ to find the other factors.

Dividing $2x^3 - 5x^2 + x + 2$ by $(x - 1)$ :

Divide the first term: $2x^3 ig/ x = 2x^2$ .
Multiply $(x - 1)$ by $2x^2$ : $2x^3 - 2x^2$ .
Subtract: $( - 5x^2 + 2x^2) + x + 2 = -3x^2 + x + 2$ .
Repeat: Divide $-3x^2 ig/ x = -3x$ ; multiply and subtract.
Finally, this gives $2$ .

Thus, we get the factorization:

$f(x) = (x - 1)(2x^2 - 3x - 2)$

Now, we can factor $2x^2 - 3x - 2$ further:

$2x^2 - 3x - 2 = (2x + 1)(x - 2)$

Thus, $f(x)$ can be factored as:

$f(x) = (x - 1)(2x + 1)(x - 2)$

Setting $f(x) = 0$ gives:

$x - 1 = 0 ightarrow x = 1$ .
$2x + 1 = 0 ightarrow x = - rac{1}{2}$ .
$x - 2 = 0 ightarrow x = 2$ .

Therefore, the solutions to the equation $f(x) = 0$ are:

$x = 1, \, x = - rac{1}{2}, \, x = 2.$

(a) Show that $(x - 1)$ is a factor of $f(x) = 2x^3 - 5x^2 + x + 2$ - Scottish Highers Maths - Question 2 - 2017

Worked Solution & Example Answer:(a) Show that $(x - 1)$ is a factor of $f(x) = 2x^3 - 5x^2 + x + 2$ - Scottish Highers Maths - Question 2 - 2017

(a) Show that $(x - 1)$ is a factor of $f(x) = 2x^3 - 5x^2 + x + 2$.

(b) Hence, or otherwise, solve $f(x) = 0$.

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