Sure! Let's factor the polynomial expression [tex]\( d^2 - 3d - 54 \)[/tex] step-by-step.
1. Write down the polynomial:
[tex]\[
d^2 - 3d - 54
\][/tex]
2. Look for two numbers that multiply to the constant term (-54) and add up to the coefficient of the linear term (-3):
We're looking for two numbers, say [tex]\(a\)[/tex] and [tex]\(b\)[/tex], such that:
[tex]\[
a \cdot b = -54
\][/tex]
and
[tex]\[
a + b = -3
\][/tex]
3. Find the pairs of factors:
Consider the possible pairs of factors of -54:
[tex]\[
\begin{align*}
(1, -54) & \quad \text{(Sum: 1 + (-54) = -53)} \\
(-1, 54) & \quad \text{(Sum: -1 + 54 = 53)} \\
(2, -27) & \quad \text{(Sum: 2 + (-27) = -25)} \\
(-2, 27) & \quad \text{(Sum: -2 + 27 = 25)} \\
(3, -18) & \quad \text{(Sum: 3 + (-18) = -15)} \\
(-3, 18) & \quad \text{(Sum: -3 + 18 = 15)} \\
(6, -9) & \quad \text{(Sum: 6 + (-9) = -3)}
\end{align*}
\][/tex]
Notice that the pair [tex]\((6, -9)\)[/tex] sums to -3.
4. Rewrite the middle term using the found pair:
The quadratic can be rewritten as:
[tex]\[
d^2 - 9d + 6d - 54
\][/tex]
5. Factor by grouping:
Group the terms into two pairs:
[tex]\[
(d^2 - 9d) + (6d - 54)
\][/tex]
Factor out the greatest common factor (GCF) from each group:
[tex]\[
d(d - 9) + 6(d - 9)
\][/tex]
6. Factor out the common binomial factor:
Since both terms have a common factor of [tex]\((d - 9)\)[/tex]:
[tex]\[
(d - 9)(d + 6)
\][/tex]
So, the factored form of [tex]\( d^2 - 3d - 54 \)[/tex] is:
[tex]\[
(d - 9)(d + 6)
\][/tex]
