Question

a ______ is an expression with two terms
a ______ is an expression with three terms
expand using ____ and collect ____ terms
when factoring always look for a ______ first
a difference of squares always has a ____ sign between two ____ terms
when factoring trinomials of the form (x^2 + bx + c), choose two numbers that ____ to give the last number and ____ to give the middle number
expanding ((a + b)^2) gives ______
expanding ((a + b)(a - b)) gives ______
factoring (a^2 - b^2) gives ______

Explanation:

Let's fill in the blanks one by one based on algebraic concepts:

Step 1: Expression with two terms

A binomial is an expression with two terms.

Step 2: Expression with three terms

A trinomial is an expression with three terms.

Step 3: Expand using __ and collect __ terms

Expand using distributive property (like FOIL for binomials) and collect like terms.

Step 4: When factoring always look for a ____ first

When factoring always look for a greatest common factor (GCF) first.

Step 5: A difference of squares always has a __ sign between two __ terms

A difference of squares always has a subtraction (-) sign between two square terms. The form is \(a^2 - b^2\).

Step 6: Factoring trinomials of the form \(x^2 + bx + c\)

When factoring trinomials of the form \(x^2 + bx + c\), choose two numbers that multiply to give the last number (\(c\)) and add to give the middle number (\(b\)).

Step 7: Expanding \((a + b)^2\)

Expanding \((a + b)^2\) gives \(a^2 + 2ab + b^2\) (using the formula \((a + b)^2 = a^2 + 2ab + b^2\)).

Step 8: Expanding \((a + b)(a - b)\)

Expanding \((a + b)(a - b)\) gives \(a^2 - b^2\) (this is the difference of squares formula).

Step 9: Factoring \(a^2 - b^2\)

Factoring \(a^2 - b^2\) gives \((a + b)(a - b)\) (since it's a difference of squares, we factor it into the product of a sum and a difference of the square roots).

Answer:

• A \(\boldsymbol{\text{binomial}}\) is an expression with two terms.
• A \(\boldsymbol{\text{trinomial}}\) is an expression with three terms.
• Expand using \(\boldsymbol{\text{distributive property}}\) and collect \(\boldsymbol{\text{like}}\) terms.
• When factoring always look for a \(\boldsymbol{\text{GCF (Greatest Common Factor)}}\) first.
• A difference of squares always has a \(\boldsymbol{\text{subtraction (-)}}\) sign between two \(\boldsymbol{\text{square}}\) terms.
• When factoring trinomials of the form \(x^2 + bx + c\), choose two numbers that \(\boldsymbol{\text{multiply}}\) to give the last number and \(\boldsymbol{\text{add}}\) to give the middle number.
• Expanding \((a + b)^2\) gives \(\boldsymbol{a^2 + 2ab + b^2}\).
• Expanding \((a + b)(a - b)\) gives \(\boldsymbol{a^2 - b^2}\).
• Factoring \(a^2 - b^2\) gives \(\boldsymbol{(a + b)(a - b)}\).