Question

dec 8 - dec 14
must post first.
when factoring a trinomial ( x^2 + 5x + 6 ), student as answer is ( (x + 2)(x + 3) ) and student bs answer is ( (x + 3)(x + 2) ). are both answers correct? explain your reasoning.
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Explanation:

Step1: Recall the distributive property (FOIL method) for multiplying binomials. The formula is \((a + b)(c + d)=ac + ad + bc + bd\).

Step2: Apply FOIL to Student A's answer \((x + 2)(x + 3)\).

First terms: \(x\times x = x^{2}\)
Outer terms: \(x\times3 = 3x\)
Inner terms: \(2\times x = 2x\)
Last terms: \(2\times3 = 6\)
Combine like terms: \(x^{2}+3x + 2x+6=x^{2}+5x + 6\), which matches the original trinomial.

Step3: Apply FOIL to Student B's answer \((x + 3)(x + 2)\).

First terms: \(x\times x = x^{2}\)
Outer terms: \(x\times2 = 2x\)
Inner terms: \(3\times x = 3x\)
Last terms: \(3\times2 = 6\)
Combine like terms: \(x^{2}+2x + 3x+6=x^{2}+5x + 6\), which also matches the original trinomial.

Step4: Recall the commutative property of multiplication, which states that \(ab = ba\) for real numbers \(a\) and \(b\). In the context of binomial multiplication, \((x + 2)(x + 3)=(x + 3)(x + 2)\) because multiplication of binomials (and numbers) is commutative.

Answer:

Both Student A's answer \((x + 2)(x + 3)\) and Student B's answer \((x + 3)(x + 2)\) are correct. This is because when we use the FOIL method to expand both binomial products, we get the original trinomial \(x^{2}+5x + 6\), and the commutative property of multiplication tells us that the order of multiplying the two binomials does not change the result.