Chapter 5

Fill in the blanks. Expressions of the form \((x+y)^{2},(x-y)^{2},\) and \((x+y)(x-y)\) occur so frequently in algebra that they are called special _____.

Fill in the blanks. \(\left(2 x^{3}\right)\left(3 x^{4}\right)\) is the product of two _____ and \((2 a-4)(3 a+5)\) is the product of two _____.

\(\left(b^{3}-b^{2}-9 b+1\right)+\left(b^{3}-b^{2}-9 b+1\right)\) is the sum of two ________.

Fill in the blanks. A ____ is a term or a sum of terms in which all variables have whole-number exponents and no variable appears in a denominator.

Fill in the blanks. \(4.84 \times 10^{5}\) is written in __________ notation. \(484,000\) is written in ___________ notation.

Fill in the blanks. In the expression \(5^{-1},\) the exponent is a _____ integer.

Fill in the blank. Expressions such as \(x^{4}, 10^{3},\) and \((5 t)^{2}\) are called _____ expressions.

Fill in the blanks. \((2 x+3)^{2}\) is the _____ of a binomial and \((a+6)(a-6)\) is the product of the sum and difference of the _____ two terms.

Fill in the blanks. We read \((x+7)(2 x-3)\) as "the _____ of \(x+7\) _____ the quantity of \(2 x-3\) "

\(\left(b^{2}-9 b+11\right)-\left(4 b^{2}-14 b\right)\) is the _______ of a trinomial and a binomial.

Fill in the blanks. The ____ of a polynomial are separated by \(+\) symbols.

Fill in the blanks. \(10^{3}, 10^{50},\) and \(10^{-4}\) are ________ of 10

Fill in the blanks. \(x^{-n}\) is the _____ of \(x^{n}\)

The expression \(\frac{x^{2}-8 x+12}{x-6}\) is a trinomial divided by _________.

Fill in the blanks to describe each special product. a. \((x+y)^{2}=x^{2}+2 x y+y^{2}\) The _____ of the second term _____ the product of the first and second terms. The square of the _____ term. b. \((x+y)(x-y)=x^{2}-y^{2}\) The square of the _____ term. The _____ of the first term.

Fill in the blanks. \(x^{3}-6 x^{2}+9 x-2\) is a polynomial in _____ variable, and is written in _____ powers of \(x,\) and \(c^{3}+2 c^{2} d-d^{2}\) is a polynomial in _____ variables and is written in _____ powers of \(d\).

Fill in the blanks. When we multiply a decimal by \(10^{5},\) the decimal point moves 5 places to the __________When we multiply a decimal by \(10^{-7}\), the decimal point moves 7 places to the _________.

Fill in the blanks. We read \(a^{0}\) as "a to the ______ power:

Fill in the blank. a. \((3 x)^{4}=\) b. \((-5 y)(-5 y)(-5 y)=\)

Consider the binomial \(5 x+4\) a. What is the square of its first term? b. What is twice the product of its two terms? c. What is the square of its second term?

Fill in the blanks. \((2 a-4)\left(3 a^{2}+5 a-1\right)\) is the product of a _____ and a _____.

The polynomial \(2 t^{4}+3 t^{3}-4 t^{2}+5 t-6\) is written in _______ powers of \(t\)

Fill in the blanks. For the polynomial \(6 x^{2}+3 x-1,\) the _____ term is \(6 x^{2},\) and the leading ____ is \(6 .\) The ____ term is \(-1\).

Fill in the blanks. Describe the procedure for converting a number from scientific notation to standard form. a. If the exponent on the base of 10 is positive, move the decimal point the same number of places to the _______ as the exponent. b. If the exponent on the base of 10 is negative, move the decimal point the same number of places to the _______ as the absolute value of the exponent.

Fill in the blank. A. \(x=x\) B. \(x^{m} x^{n}=\) C. \((x y)^{n}=\) D. \(\left(a^{b}\right)^{c}=\) E. \(\frac{x^{m}}{x^{n}}=\) F. \(\left(\frac{a}{b}\right)^{n}=\)

Fill in the blanks. We read \(3^{-4}\) as " 3 to the ______ ______ power:

The long division method is a series of four steps that are repeated. Put them in the correct order: subtract \(\quad\) multiply \(\quad\) bring down \(\quad\) divide

Complete each solution to find the product. $$ \begin{aligned} (x+4)^{2} &=\square^{2}+2(x)(\square)+\square^{2} \\ &=x^{2}+\square+16 \end{aligned} $$

Fill in the blanks. A. To multiply two polynomials, multiply _____ term of one polynomial by _____ term of the other polynomial, and then combine like terms. B. When multiplying three polynomials, we begin by multiplying _____ two of them, and then we multiply that result by the _____ polynomial.

Fill in the blanks. A ____ is a polynomial with exactly one term. A ____ is a polynomial with exactly two terms. A ____ is a polynomial with exactly three terms.

Fill in the blanks. a. When a real number greater than or equal to 10 is written in scientific notation, the exponent on 10 is a _______ integer. b. When a real number between 0 and 1 is written in scientific notation, the exponent on 10 is a _______ integer.

Fill in the blank. To simplify each expression, determine whether you add, subtract, multiply, or divide the exponents. A. \(\frac{x^{8}}{x^{2}}\) B. \(b^{6} \cdot b^{9}\) C. \(\left(n^{8}\right)^{4}\) D. \(\left(a^{4} b^{2}\right)^{5}\)

Fill in the blanks. The ____ of the term \(3 x^{7}\) is 7 because \(x\) appears as a factor 7 times: \(3 \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x\).

Fill in the blanks. The arrows show the movement of a decimal point. By what power of 10 was each decimal multiplied? a. 0.0000000556 b. \(8,04,1,000,000\)

Fill in the blank. A. To simplify \(\left(2 y^{3} z^{2}\right)^{4},\) what factors within the parentheses must be raised to the fourth power? B. To simplify \(\left(\frac{y^{3}}{z^{2}}\right)^{4},\) what two expressions must be raised to the fourth power?

Complete each rule for exponents. \(\begin{array}{ll}{\text { a. } x^{m} \cdot x^{n}=} & {\text { b. } x^{0}=} \\\ {\text { c. }\left(x^{m}\right)^{n}=} & {\text { d. }(x y)^{n}=} \\\ {\text { e. }\left(\frac{x}{y}\right)^{n}=} & {\text { f. } x^{-n}=} \\\ {\text { g. } \frac{1}{x^{-n}}=} & {\text { h. } \frac{x^{-m}}{y^{-n}}=} \\\ {\text { i. } \frac{x^{m}}{x^{n}}=} & {\text { j. }\left(\frac{x}{y}\right)^{-n}=}\end{array}\)

Fill in the blanks: To check an answer of a long division, we use the fact that Divisor _________ + remainder =

Complete each solution to find the product. $$ \begin{aligned} (s+5)(s-5) &=\square^{2}-\square^{2} \\ &=s^{2}-\square \end{aligned} $$

Simplify each polynomial by combining like terms. a. \(6 x^{2}-8 x+9 x-12\) b. \(5 x^{4}+3 a x^{2}+5 a x^{2}+3 a^{2}\)

Simplify each polynomial, if possible. a. \(2 x^{2}+3 x^{2}\) b. \(15 m^{3}-m^{3}\) c. \(8 a^{3} b-a^{3} b\) d. \(6 c d+4 c^{2} d\)

To ____ the polynomial \(x^{2}-2 x+1\) for \(x=6,\) we substitute 6 for \(x\) and follow the rules for the order of operations.

Simplify each expression, if possible. A. \(x^{2}+x^{2}\) B. \(x^{2} \cdot x^{2}\) C. \(x^{2}+x\) D. \(x^{2} \cdot x\)

Complete each table. \(\begin{array}{|r|r|}\hline x & {3^{x}} \\ \hline 2 & {} \\ \hline 1 & {} \\\ \hline 0 & {} \\ \hline-1 & {} \\ \hline-2 & {} \\ \hline\end{array}\)

Check to see whether the following result of a long division is correct. $$ \frac{x^{2}+4 x-20}{x-3}=x+7+\frac{1}{x-3} $$

True or false: \((t+7)(t-7)=(t-7)(t+7) ?\)

\((3 a)\left(2 a^{2}\right)\) can be classified as a monomial - monomial. Classify the following products by identifying the types of polynomial factors. a. \(6 x(x-7)\) b. \((9 a+1)(5 a-3)\) c. \((c-d)\left(c^{2}-c+d\right)\) d. \(6 m\left(m^{2}+1\right)\left(m^{2}-1\right)\)

What is the result of the addition in the \(x\) -column? $$ \begin{array}{l} {4 x^{2}+x-12} \\ {5 x^{2}-8 x+23} \\ \hline \end{array} $$

Fill in the blanks to write number in scientific notation. a. \(0.0082=\quad \times 10^{-3}\) b. \(0.0000001=\quad \times 10^{-7}\) c. \(0.00003457=3.457 \times 10\)

Fill in the blanks. The graph of \(y=x^{2}\) is a cup-shaped curve called a ____.

Simplify each expression, if possible. A. \(x^{3}-x^{2}\) B. \(\frac{x^{3}}{x^{2}}\) C. \(4^{2} \cdot 2^{4}\) D. \(\frac{x^{3}}{y^{2}}\)