Question

factor the quadratic.
$z^{2}+6z - 7$
convert from general form $ax^{2}+bx + c$ to factored form $(kx + m)(hx + n)$. for additional help, use the area model below to help you determine the correct length and width to represent the trinomial $z^{2}+6z - 7$.

• determine the product of the a and c terms (which already appear inside the factor table).

product of a and c terms: $-7z^{2}$

• in the other inner corners of the factor table, enter two factors of $-7z^{2}$ whose sum equals the b term, $6z$.
• in each outer cell of the factor table, enter the greatest common factor (gcf) of the two inner cell quantities in that row or column.

Explanation:

Step1: Identify coefficients

For the quadratic $z^{2}+6z - 7$, $a = 1$, $b=6$, $c=-7$. The product $ac=1\times(-7)=-7$.

Step2: Find two - numbers

We need to find two numbers that multiply to $-7$ and add up to $6$. The numbers are $7$ and $- 1$ since $7\times(-1)=-7$ and $7+( - 1)=6$.

Step3: Rewrite the middle term

Rewrite $z^{2}+6z - 7$ as $z^{2}+7z - z - 7$.

Step4: Group the terms

Group the terms: $(z^{2}+7z)+(-z - 7)$.

Step5: Factor out GCF from each group

Factor out the GCF from each group. From $z^{2}+7z$, the GCF is $z$, so we get $z(z + 7)$. From $-z - 7$, the GCF is $-1$, so we get $-1(z + 7)$.

Step6: Factor out the common binomial

Factor out the common binomial $(z + 7)$: $(z + 7)(z-1)$.

Answer:

$(z + 7)(z - 1)$