Question4 $\qquad$ olifie les expressions suivantes.

Studdy Solution

STEP 1

What is this asking? We need to simplify the expression $\frac{8x^2 - 2x - 1}{3x^2 + x + 7}$ , if possible! Watch out! We can't just "cancel" the $x^2$ terms!
We need to see if the numerator and denominator share any common factors.

STEP 2

1. Factor the numerator
2. Factor the denominator
3. Simplify the expression

STEP 3

Our numerator is $8x^2 - 2x - 1$ .
Let's factor this bad boy!

STEP 4

We're looking for two numbers that multiply to $8 \cdot -1 = -8$ and add up to $-2$ .
Those numbers are $-4$ and $+2$ .

STEP 5

We rewrite our numerator as $8x^2 - 4x + 2x - 1$ .

STEP 6

Now, we can factor by grouping: $4x(2x - 1) + 1(2x - 1)$ .

STEP 7

This gives us $(4x + 1)(2x - 1)$ .
Awesome!

STEP 8

Our denominator is $3x^2 + x + 7$ .

STEP 9

We are looking for two numbers that multiply to $3 \cdot 7 = 21$ and add up to $1$ .
Hmm, this seems tricky...

STEP 10

There aren't any integer factors that work!
This means our denominator is prime, so we can't factor it any further.

STEP 11

Our original expression is now $\frac{(4x + 1)(2x - 1)}{3x^2 + x + 7}$ .

STEP 12

Since the numerator and denominator don't share any common factors, we can't simplify this expression any further.
That means we're done!

STEP 13

The simplified expression is $\frac{8x^2 - 2x - 1}{3x^2 + x + 7}$ , which is the same as what we started with!
Sometimes, expressions are already in their simplest form.

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