QuestionSolve by factoring:
Studdy Solution
STEP 1
1. The expression is a quadratic polynomial in terms of .
2. The goal is to factor the quadratic expression into two binomials.
3. The solution involves finding two numbers that multiply to the constant term and add to the linear coefficient.
STEP 2
1. Identify the coefficients of the quadratic expression.
2. Determine two numbers that multiply to the constant term and add to the linear coefficient.
3. Use these numbers to factor the quadratic expression.
4. Verify the factorization by expanding the binomials.
STEP 3
Identify the coefficients of the quadratic expression :
- The coefficient of is .
- The coefficient of is .
- The constant term is .
STEP 4
Determine two numbers that multiply to the constant term and add to the linear coefficient .
- We need two numbers and such that:
$ a \cdot b = -14 \quad \text{and} \quad a + b = -5
\]
STEP 5
Find the pair of numbers:
- Consider the factors of : .
- The pair that satisfies both conditions is because:
$ 2 \cdot (-7) = -14 \quad \text{and} \quad 2 + (-7) = -5
\]
STEP 6
Use these numbers to factor the quadratic expression:
- Rewrite the middle term using the numbers and :
$ p^2 - 5p - 14 = p^2 + 2p - 7p - 14
\]
STEP 7
Factor by grouping:
- Group the terms: and .
- Factor out the greatest common factor from each group:
$ p(p + 2) - 7(p + 2)
\]
STEP 8
Factor out the common binomial factor:
- The common factor is :
$ (p - 7)(p + 2)
\]
STEP 9
Verify the factorization by expanding the binomials:
- Expand :
$ p^2 + 2p - 7p - 14 = p^2 - 5p - 14
\]
- The expansion matches the original expression, confirming the factorization is correct.
The factored form of the quadratic expression is:
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