- Given the expression \(6x^2-9(x-4)+2\), which of the following represents a coefficient?
- 6
- 9
- –4
- 2
A coefficient is the number in front of any term containing a variable or variables. In this case, 6 is the coefficient of \(6x^2\).
- How many terms are included in the following expression?
- 3
- 4
- 5
- 6
A term is a part of an expression that may include a number and/or variable(s) that is separated from other parts of the expression by the operations of addition and/or subtraction. In the given expression, there are four terms: \(2x^2\), \(12x\), \(8\), and \(10x\).
- Which of the following represents a factor in the formula below?
- \(2\pi r\)
- \(\pi r^2\)
- \(l+r\)
- \(\pi rl\)
A factor of an algebraic equation is similar to a factor in a multiplication problem; it is a quantity that is multiplied by another quantity to give a product. In this case, the quantity \((l + r)\) is multiplied by the quantity \(\pi r\). Thus, \((l + r)\) is a factor in the given formula.
- Which of the following expressions is equivalent to \((x-3)^2\) ?
- \(x^2-3x+9\)
- \(x^2-6x-9\)
- \(x^2-6x+9\)
- \(x^2+3x-9\)
The expression can be written as \((x-3)(x-3)\). Distribution gives \(x^2-3x-3x+9\). Combining like terms gives \(x^2-6x+9\).
- Which of the following represents the factors of the expression \(x^2-3x-40\) ?
- \((x-8)(x+5)\)
- \((x-7)(x+4)\)
- \((x+10)(x-4)\)
- \((x+6)(x-9)\)
The expression may be factored as \((x-8)(x+5)\). The factorization may be checked by distributing each term in the first factor over each term in the second factor. Doing so gives \(x^2+5x-8x-40\), which can be rewritten as \(x^2-3x-40\).
- Which of the following represents the zeros of the expression \(x^2-2x-24\) ?
- \(x=4\) and \(x=6\)
- \(x=-4\) and \(x=6\)
- \(x=4\) and \(x=-6\)
- \(x=-4\) and \(x=-6\)
The quadratic expression may be factored as \((x-6)(x+4)\). Setting each factor equal to 0 gives \(x-6=0\) and \(x+4=0\). Solving for \(x\) gives \(x=6\) and \(x=-4\).
- What are the zeros of a quadratic expression, represented by the factors \(x+6\) and \(x-7\) ?
- \(x=6\) and \(x=7\)
- \(x=6\) and \(x=-7\)
- \(x=-6\) and \(x=7\)
- \(x=-6\) and \(x=-7\)
The zeros of an expression are the points at which the corresponding \(y\)-values are 0. Thus, the zeros of the expression, represented by the given factors, will occur at the \(x\)-values that have corresponding \(y\)-values of 0.
Setting each factor equal to 0 gives \(x+6=0\) and \(x-7=0\). Solving for \(x\) gives \(x=-6\) and \(x=7\). Thus, the zeros of the expression are \(x=-6\) and \(x=7\).
- What is the minimum value of the expression \(3x^2-6x+6\) ?
- –3
- 0
- 3
- 6
To find the minimum, complete the square:
\(3x^2 – 6x + 6 = 3\bigl(x^2 – 2x\bigr) + 6\) \(= 3\bigl((x – 1)^2 – 1\bigr) + 6 = 3(x – 1)^2 – 3 + 6\) \(= 3(x – 1)^2 + 3\)
Since \((x – 1)^2 \ge 0\) for all real \(x\), the smallest value of \(3(x – 1)^2\) is 0 (when \(x = 1\)). Thus, the minimum of the entire expression is \(0 + 3 = 3\).
- Which of the following expressions is equivalent to \(\frac{5}{8}^3\) ?
- \(5\frac{3}{8}^3\)
- \(5\frac{3}{8}\)
- \(\frac{5}{8^3}\)
- \(\frac{8}{5}^{\frac{1}{3}}\)
One of the properties of exponents states the following:
\(\frac{a}{b}^n=\frac{an}{b}^n\)
Therefore, the given expression can also be written as \(5\frac{3}{8}^3\).
- Kevin saves two dollars during the month of January. Each month, he plans to save twice the amount saved during the previous month. With this plan, how much will he have saved after 18 months?
- $218,174
- $262,144
- $478,195
- $524,286
Each month’s savings doubles:
- Month 1: \($2=2^1\)
- Month 2: \($4=2^2\)
- \(…\)
- Month 18: \($2^{18}\)
The total after 18 months is the sum of a geometric series:
\(S = 2 + 2^2 + \cdots + 2^{18}\) \(= \sum_{k=1}^{18} 2^k = 2\cdot\frac{2^{18}-1}{2-1}\) \(= 2^{19} – 2 = 524{,}288 – 2\) \(= 524{,}286\)
Thus, he will have saved $524,286.