Question

1. now that we have studied some tools for factoring:

a) reflect: what does it mean to factor an expression? what is the goal of factoring?
b) summarize the methods for factoring by grouping and for factoring trinomials (expressions with 3 terms) that we have studied so far. explain/summarize as if you were teaching a peer.

Explanation:

For part a), the concept of factoring and its goals are fundamental in algebra. For part b), factoring by grouping and factoring trinomials are key algebraic techniques. Factoring by grouping uses the distributive property in reverse, and factoring trinomials involves finding appropriate numbers to rewrite and factor the expression.

Answer:

a) Factoring an expression means rewriting it as a product of simpler expressions. The goal of factoring is to simplify algebraic expressions, solve equations (by setting factors equal to zero), and understand the structure of polynomials.
b) Factoring by grouping:

• For an expression with four terms like \(ax + ay+bx + by\), group the terms in pairs. So, \((ax + ay)+(bx + by)\).
• Factor out the greatest - common factor from each pair. From the first pair, we get \(a(x + y)\) and from the second pair, we get \(b(x + y)\).
• Then, factor out the common binomial factor \((x + y)\) to get \((a + b)(x + y)\).
• Factoring trinomials (\(ax^{2}+bx + c\)):
• If \(a = 1\) (e.g., \(x^{2}+bx + c\)), we need to find two numbers \(m\) and \(n\) such that \(m\times n=c\) and \(m + n=b\). Then we can rewrite the middle term as \(x^{2}+mx+nx + c\) and factor by grouping. For example, for \(x^{2}+5x + 6\), we find \(m = 2\) and \(n = 3\) since \(2\times3 = 6\) and \(2+3 = 5\). So \(x^{2}+5x + 6=x^{2}+2x+3x + 6=x(x + 2)+3(x + 2)=(x + 2)(x + 3)\).
• If \(a

eq1\) (e.g., \(3x^{2}+5x+2\)), we multiply \(a\times c\) (here \(3\times2 = 6\)). Then find two numbers that multiply to \(a\times c\) and add up to \(b\). For \(3x^{2}+5x + 2\), we find \(m = 2\) and \(n = 3\) since \(2\times3=6\) and \(2 + 3=5\). Rewrite the middle term: \(3x^{2}+2x+3x + 2=x(3x + 2)+1(3x + 2)=(3x + 2)(x + 1)\).