Question

1. (challenge) factor by grouping, and check with distribution. $x^{2}-6x + 6d - dx$
2. look carefully at the answers you have for question (4.) and question (5.) and use a similar end result to create an example of an expression that can be factored by grouping. then, factor it by grouping.
3. factor each of the following, if possible. check with distribution.

a) $y^{2}+4y - 21$
b) $y^{2}+4y + 21$
c) $2w^{2}-8w + 42$

Explanation:

Question 5:

Step1: Group the terms

$(x^{2}-6x)+(6d - dx)$

Step2: Factor out the common factors from each group

$x(x - 6)-d(x - 6)$

Step3: Factor out the common binomial factor

$(x - d)(x - 6)$

Step4: Check with distribution

$(x - d)(x - 6)=x^{2}-6x - dx+6d=x^{2}-6x + 6d - dx$

Question 7a:

Step1: Find two numbers that multiply to $-21$ and add to $4$

The numbers are $7$ and $-3$ since $7\times(-3)=-21$ and $7+( - 3)=4$

Step2: Rewrite the middle term

$y^{2}+7y-3y - 21$

Step3: Group the terms

$(y^{2}+7y)-(3y + 21)$

Step4: Factor out the common factors from each group

$y(y + 7)-3(y + 7)$

Step5: Factor out the common binomial factor

$(y + 7)(y - 3)$

Step6: Check with distribution

$(y + 7)(y - 3)=y^{2}-3y+7y - 21=y^{2}+4y - 21$

Question 7b:

For the quadratic $y^{2}+4y + 21$, the discriminant $\Delta=b^{2}-4ac$ where $a = 1$, $b = 4$, and $c = 21$.
$\Delta=4^{2}-4\times1\times21=16-84=-68<0$. So it cannot be factored over the real - numbers.

Question 7c:

Step1: Factor out the greatest common factor

$2(w^{2}-4w + 21)$
The quadratic $w^{2}-4w + 21$ has discriminant $\Delta=(-4)^{2}-4\times1\times21=16 - 84=-68<0$, so we leave it as $2(w^{2}-4w + 21)$ over the real - numbers.

Answer:

Question 5:

$(x - d)(x - 6)$

Question 7a:

$(y + 7)(y - 3)$

Question 7b:

Cannot be factored over the real - numbers

Question 7c:

$2(w^{2}-4w + 21)$