Special Products of Binomials Miss Calo

Special Products For this activity, we will be calling the first term of a binomial “a” and the second term of a binomial “b.” For example, in (x + 5), x would be “a” and 5 would be “b.” In (x – 1), what is “a” and what is “b”?

Investigation As you work through each question, record the x2 term, the x term, and the constant term. Look for a pattern between the terms in the question and the terms in the solution. In the last row, see if you can find a “short cut” to the solution without having to perform all the math steps. Remember, the first term in the binomial is represented by “a,” and the second term is represented by “b.”

FOIL it! (x + 1)(x + 1) (x + 2)(x + 2)

Helpful Questions if you’re stuck What patterns do you see in the table? What do you notice about the first terms of each solution? What does that tell us about what happens to “a” in problems like these? Can you think of a shortcut to get to the x term quickly in these problems without FOILing it out? What do you notice about the constant in each row? How can we quickly get the constant value, looking at our values for “a” and “b”?

Let’s check our rules (a + b)2 = (a + b)(a + b) =

Our rule When you square a binomial with addition between its terms, like (a + b), you square the first term in the binomial, add it to the product of the first and second term doubled, and add it to the second term squared.

Our rule for binomials in the form of (a + b)2 So, we know when we see a binomial multiplied by itself with addition in the middle the binomials, we can use this shortcut. Will this shortcut work for problems such as these? (x + 1)(x + 2) (x + 1)(x – 1) Now, we are going to work on a shortcut for multiplying a binomial by itself with subtraction in the middle of the binomial.

FOIL It! (x – 1)(x – 1) (x – 2)(x – 2)

Helpful hints What patterns do you see in the table? What do you notice about the first terms of each solution? What does that tell us about what happens to “a” in problems like these? Can you think of a shortcut to get to the x term quickly in these problems without FOILing it out? What do you notice about the constant in each row? How can we quickly get the constant value, looking at our values for “a” and “b”?

Let’s check our rules (a – b)2 = (a – b)(a – b) =

Our rule for binomials in the form of (a – b)2 When multiplying a binomial by itself with subtraction between its terms, you square the first term, subtract the product of the first and second term doubled, and add it to the second term squared. So, we know when we see a binomial with subtraction between its terms multiplied by itself, we can use this shortcut. We won’t have to FOIL it!

Foil it! (x + 1)(x – 1) (x + 2)(x – 2)

Helpful hints What patterns do you see in the table? What do you notice about the first terms of each solution? What does that tell us about what happens to “a” in problems like these? Can you think of a shortcut to get to the x term quickly in these problems without FOILing it out? What do you notice about the constant in each row? How can we quickly get the constant value, looking at our values for “a” and “b”?

Let’s check our rules (a – b)(a + b) =

Our rule for binomials in the form of (a – b)(a + b) When multiplying a binomial in the form of (a – b)(a + b), you square the first term and subtract the second term squared.

Kahoot?

assessment Miss Calo has a square yard. The length and width of her yard are both represented by the polynomial (x + 5). She also has a rectangular pool in her yard. The width of the pool is represented by the polynomial (x – 2), and the length of the pool is represented by the polynomial (x + 2). Write a polynomial that represents the area of the yard AROUND THE POOL. Hint: Draw a picture!