문자와 식 전체 문제

Mental math You can use the product of conjugates pattern to multiply numbers without a calculator. Say you need to multiply 47 times 53. Think of 47 as $50-3$ and 53 as $50+3$.

• ⓐ Multiply $(50-3)(50+3)$ by using the product of conjugates pattern, $(a-b)(a+b)={a}^{2}-{b}^{2}$.
• ⓑ Multiply $47\cdot 53$ without using a calculator.
• ⓒ Which way is easier for you? Why?

Mental math You can use the binomial squares pattern to multiply numbers without a calculator. Say you need to square 65. Think of 65 as $60+5$.

• ⓐ Multiply ${(60+5)}^{2}$ by using the binomial squares pattern, ${(a+b)}^{2}={a}^{2}+2ab+{b}^{2}$.
• ⓑ Square 65 without using a calculator.
• ⓒ Which way is easier for you? Why?

How do you decide which pattern to use?

Why does ${(a+b)}^{2}$ result in a trinomial, but $(a-b)(a+b)$ result in a binomial?

Marta did the following work on her homework paper:

$$\begin{array}{c}{(3-y)}^{2} \\ \,{3}^{2}-{y}^{2} \\ \quad 9-{y}^{2}\end{array}$$

Explain what is wrong with Marta’s work.

Use the order of operations to show that ${(3+5)}^{2}$ is 64, and then use that numerical example to explain why ${(a+b)}^{2}\ne {a}^{2}+{b}^{2}$.

Identify Polynomials, Monomials, Binomials and Trinomials

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

ⓐ $11{c}^{4}-23{c}^{2}+1$

ⓑ $9{p}^{3}+6{p}^{2}-p-5$

ⓒ $\frac{3}{7}x+\frac{5}{14}$

ⓓ 10

ⓔ $2y-12$

Identify Polynomials, Monomials, Binomials and Trinomials

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

ⓐ ${a}^{2}-{b}^{2}$

ⓑ $24{d}^{3}$

ⓒ ${x}^{2}+8x-10$

ⓓ ${m}^{2}{n}^{2}-2mn+6$

ⓔ $7{y}^{3}+{y}^{2}-2y-4$

Determine the Degree of Polynomials

In the following exercises, determine the degree of each polynomial.

• ⓐ $3{x}^{2}+9x+10$
• ⓑ $14{a}^{2}bc$
• ⓒ $6y+1$
• ⓓ ${n}^{3}-4{n}^{2}+2n-8$
• ⓔ $-19$

Determine the Degree of Polynomials

In the following exercises, determine the degree of each polynomial.

• ⓐ $5{p}^{3}-8{p}^{2}+10p-4$
• ⓑ $-20{q}^{4}$
• ⓒ ${x}^{2}+6x+12$
• ⓓ $23{r}^{2}{s}^{2}-4rs+5$
• ⓔ 100

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

${\,\text{5y}}^{3}+8{y}^{3}$

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$-14k+19k$

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$12q-(-6q)$

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$-9c-18c$

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$\text{12x}-4y-9x$

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$3{m}^{2}+7{n}^{2}-3{m}^{2}$

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$6{x}^{2}y-4x+8x{y}^{2}$

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$\text{13a}+b$

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

$(5{x}^{2}+12x+1)+(6{x}^{2}-8x+3)$

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

$(9{p}^{2}-5p+3)+(4{p}^{2}-4)$