Stewart Precalc 6e Section 1.3: Algebraic Expressions

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Stewart Precalc 6e Section 1.3: Algebraic Expressions 0/64
1 Exercise - Factoring by Grouping · Level 3
Factor the expression by grouping terms: \(2 x^3 + x^2 - 6 x - 3\).
2 Exercise - Factoring by Grouping · Level 3
Factor the expression by grouping terms: \(-9 x^3 - 3 x^2 + 3 x + 1\).
3 Exercise - Factoring by Grouping · Level 3
Factor the expression by grouping terms: \(x^3 + x^2 + x + 1\).
4 Exercise - Factoring by Grouping · Level 3
Factor the expression by grouping terms: \(x^5 + x^4 + x + 1\).
5 Exercise - Fractional Exponents · Level 3
Factor the expression completely by factoring out the lowest power of each common factor: \(x^{\dfrac{5}{2}} - x^{\dfrac{1}{2}}\).
6 Exercise - Fractional Exponents · Level 4
Factor the expression completely by factoring out the lowest power of each common factor: \(3 x^{-\dfrac{1}{2}} + 4 x^{\dfrac{1}{2}} + x^{\dfrac{3}{2}}\).
7 Exercise - Fractional Exponents · Level 4
Factor the expression completely by factoring out the lowest power of each common factor: \(x^{-\dfrac{3}{2}} + 2 x^{-\dfrac{1}{2}} + x^{\dfrac{1}{2}}\).
8 Exercise - Fractional Exponents · Level 4
Factor the expression completely by factoring out the lowest power of each common factor: \((x - 1)^{\dfrac{7}{2}} - (x - 1)^{\dfrac{3}{2}}\).
9 Exercise - Fractional Exponents · Level 4
Factor the expression completely by factoring out the lowest power of each common factor: \((x^2 + 1)^{\dfrac{1}{2}} + 2(x^2 + 1)^{-\dfrac{1}{2}}\).
10 Exercise - Fractional Exponents · Level 4
Factor the expression completely by factoring out the lowest power of each common factor: \(x^{-\dfrac{1}{2}}(x + 1)^{\dfrac{1}{2}} + x^{\dfrac{1}{2}}(x + 1)^{-\dfrac{1}{2}}\).
11 Exercise - Complete Factoring · Level 2
Factor the expression completely: \(12 x^3 + 18 x\).
12 Exercise - Complete Factoring · Level 2
Factor the expression completely: \(30 x^3 + 15 x^4\).
13 Exercise - Complete Factoring · Level 2
Factor the expression completely: \(x^2 - 2 x - 8\).
14 Exercise - Complete Factoring · Level 2
Factor the expression completely: \(x^2 - 14 x + 48\).
15 Exercise - Complete Factoring · Level 2
Factor the expression completely: \(2 x^2 + 5 x + 3\).
16 Exercise - Complete Factoring · Level 2
Factor the expression completely: \(2 x^2 + 7 x - 4\).
17 Exercise - Complete Factoring · Level 3
Factor the expression completely: \(9 x^2 - 36 x - 45\).
18 Exercise - Complete Factoring · Level 3
Factor the expression completely: \(8 x^2 + 10 x + 3\).
19 Exercise - Complete Factoring · Level 2
Factor the expression completely: \(49 - 4 y^2\).
20 Exercise - Complete Factoring · Level 2
Factor the expression completely: \(4 t^2 - 9 s^2\).
21 Exercise - Complete Factoring · Level 2
Factor the expression completely: \(t^2 - 6 t + 9\).
22 Exercise - Complete Factoring · Level 2
Factor the expression completely: \(x^2 + 10 x + 25\).
23 Exercise - Complete Factoring · Level 2
Factor the expression completely: \(4 x^2 + 4 x y + y^2\).
24 Exercise - Complete Factoring · Level 2
Factor the expression completely: \(r^2 - 6 r s + 9 s^2\).
25 Exercise - Complete Factoring · Level 3
Factor the expression completely: \((a + b)^2 - (a - b)^2\).
26 Exercise - Complete Factoring · Level 3
Factor the expression completely: \(\left(1 + \dfrac{1}{x}\right)^2 - \left(1 - \dfrac{1}{x}\right)^2\).
27 Exercise - Complete Factoring · Level 3
Factor the expression completely: \(x^2(x^2 - 1) - 9(x^2 - 1)\).
28 Exercise - Complete Factoring · Level 3
Factor the expression completely: \((a^2 - 1) b^2 - 4(a^2 - 1)\).
29 Exercise - Complete Factoring · Level 3
Factor the expression completely: \(8 x^3 - 125\).
30 Exercise - Complete Factoring · Level 3
Factor the expression completely: \(x^6 + 64\).
31 Exercise - Complete Factoring · Level 2
Factor the expression completely: \(x^3 + 2 x^2 + x\).
32 Exercise - Complete Factoring · Level 2
Factor the expression completely: \(3 x^3 - 27 x\).
33 Exercise - Complete Factoring · Level 3
Factor the expression completely: \(x^4 y^3 - x^2 y^5\).
34 Exercise - Complete Factoring · Level 3
Factor the expression completely: \(18 y^3 x^2 - 2 x y^4\).
35 Exercise - Complete Factoring · Level 3
Factor the expression completely: \(2 x^3 + 4 x^2 + x + 2\).
36 Exercise - Complete Factoring · Level 3
Factor the expression completely: \(3 x^3 + 5 x^2 - 6 x - 10\).
37 Exercise - Complete Factoring · Level 3
Factor the expression completely: \((x - 1)(x + 2)^2 - (x - 1)^2(x + 2)\).
38 Exercise - Complete Factoring · Level 4
Factor the expression completely: \(y^4(y + 2)^3 + y^5(y + 2)^4\).
39 Exercise - Complete Factoring · Level 4
Factor the expression completely: \((a^2 + 1)^2 - 7(a^2 + 1) + 10\).
40 Exercise - Complete Factoring · Level 4
Factor the expression completely: \((a^2 + 2 a)^2 - 2(a^2 + 2 a) - 3\).
41 Exercise - Product Rule · Level 4
Factor the expression completely (this type arises in calculus using the Product Rule): \(5(x^2 + 4)^4 (2 x)(x - 2)^4 + (x^2 + 4)^5 (4)(x - 2)^3\).
42 Exercise - Product Rule · Level 4
Factor the expression completely: \(3(2 x - 1)^2 (2)(x + 3)^{\dfrac{1}{2}} + (2 x - 1)^3 \left(\dfrac{1}{2}\right)(x + 3)^{-\dfrac{1}{2}}\).
43 Exercise - Product Rule · Level 4
Factor the expression completely: \((x^2 + 3)^{-\dfrac{1}{3}} - \left(\dfrac{2}{3}\right) x^2 (x^2 + 3)^{-\dfrac{4}{3}}\).
44 Exercise - Product Rule · Level 4
Factor the expression completely: \(\left(\dfrac{1}{2}\right) x^{-\dfrac{1}{2}}(3 x + 4)^{\dfrac{1}{2}} - \left(\dfrac{3}{2}\right) x^{\dfrac{1}{2}}(3 x + 4)^{-\dfrac{1}{2}}\).
45 Exercise - Verification · Level 4
(a) Show that \(a b = \left(\dfrac{1}{2}\right)[(a + b)^2 - (a^2 + b^2)]\).
(b) Show that \((a^2 + b^2)^2 - (a^2 - b^2)^2 = 4 a^2 b^2\).
(c) Show that \((a^2 + b^2)(c^2 + d^2) = (a c + b d)^2 + (a d - b c)^2\).
(d) Factor completely: \(4 a^2 c^2 - (a^2 - b^2 + c^2)^2\).

Enter your answer directly below each part above.

46 Exercise - Verification · Level 3
Verify Special Factoring Formulas 4 and 5 by expanding their right-hand sides: Formula 4: \(A^3 - B^3 = (A - B)(A^2 + A B + B^2)\) Formula 5: \(A^3 + B^3 = (A + B)(A^2 - A B + B^2)\)
47 Exercise - Applications · Level 3
A culvert is constructed out of large cylindrical shells cast in concrete, as shown in the figure. Using the formula for the volume of a cylinder, explain why the volume of the cylindrical shell is \( V = \pi R^2 h - \pi r^2 h \) Factor to show that \( V = 2 \pi \cdot \text{average radius} \cdot \text{height} \cdot \text{thickness} \)
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48 Exercise - Applications · Level 3
A square field is mowed around the edges every week. The field measures \(b\) feet by \(b\) feet, and the mowed strip is \(x\) feet wide (see the figure).
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(a) Explain why the area of the mowed portion is \(b^2 - (b - 2 x)^2\).
(b) Factor the expression in part (a) to show that the area of the mowed portion is also \(4 x(b - x)\).

Enter your answer directly below each part above.

49 Exercise - Discovery and Writing · Level 3
Make up several pairs of polynomials, then calculate the sum and product of each pair. On the basis of your experiments and observations, answer the following questions.
(a) How is the degree of the product related to the degrees of the original polynomials?
(b) How is the degree of the sum related to the degrees of the original polynomials?

Enter your answer directly below each part above.

50 Exercise - Discovery and Writing · Level 3
Use the Difference of Squares Formula to evaluate each expression mentally.
(a) \(528^2 - 527^2\)
(b) \(122^2 - 120^2\)
(c) \(1020^2 - 1010^2\) Now use the Special Product Formula \((A + B)(A - B) = A^2 - B^2\) to evaluate these products in your head:
(d) \(79 \cdot 51\)
(e) \(998 \cdot 1002\)

Enter your answer directly below each part above.

51 Exercise - Discovery and Writing · Level 4
(a) Factor the expressions completely: \(A^4 - B^4\) and \(A^6 - B^6\).
(b) Verify that \(18{,}335 = 12^4 - 7^4\) and that \(2{,}868{,}335 = 12^6 - 7^6\).
(c) Use the results of parts (a) and (b) to factor the integers \(18{,}335\) and \(2{,}868{,}335\). Then show that in both of these factorizations, all the factors are prime numbers.

Enter your answer directly below each part above.

52 Exercise - Discovery and Writing · Level 4
Verify these formulas by expanding and simplifying the right-hand side: \(A^2 - 1 = (A - 1)(A + 1)\) \(A^3 - 1 = (A - 1)(A^2 + A + 1)\) \(A^4 - 1 = (A - 1)(A^3 + A^2 + A + 1)\) On the basis of the pattern displayed in this list, how do you think \(A^5 - 1\) would factor? Verify your conjecture. Now generalize the pattern you have observed to obtain a factoring formula for \(A^n - 1\), where \(n\) is a positive integer.
53 Exercise - Discovery and Writing · Level 4
A trinomial of the form \(x^4 + a x^2 + b\) can sometimes be factored easily. For example, \( x^4 + 3 x^2 - 4 = (x^2 + 4)(x^2 - 1) \) But \(x^4 + 3 x^2 + 4\) cannot be factored in this way. Instead, we can use the following method: \(x^4 + 3 x^2 + 4 = (x^4 + 4 x^2 + 4) - x^2 = (x^2 + 2)^2 - x^2 = (x^2 - x + 2)(x^2 + x + 2)\). Factor the following, using whichever method is appropriate.
(a) \(x^4 + x^2 - 2\)
(b) \(x^4 + 2 x^2 + 9\)
(c) \(x^4 + 4 x^2 + 16\)
(d) \(x^4 + 2 x^2 + 1\)

Enter your answer directly below each part above.

54 Example - Multiplying Binomials Using FOIL · Level 2
Use FOIL to find the product: \((2x + 1)(3x - 5)\).
55 Example - Multiplying Polynomials · Level 2
Find the product: \((2x + 3)(x^2 - 5x + 4)\).
56 Example - Using the Special Product Formulas · Level 2
Use the Special Product Formulas to find each product. (a) \((3x + 5)^2\) (b) \((x^2 - 2)^3\)
57 Example - Using the Special Product Formulas · Level 3
Find each product. (a) \((2x - \sqrt{y})(2x + \sqrt{y})\) (b) \((x + y - 1)(x + y + 1)\)
58 Example - Factoring Out Common Factors · Level 2
Factor each expression. (a) \(3x^2 - 6x\) (b) \(8x^4 y^2 + 6x^3 y^3 - 2x y^4\) (c) \((2x + 4)(x - 3) - 5(x - 3)\)
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59 Example - Factoring x^2 + bx + c by Trial and Error · Level 2
Factor: \(x^2 + 7x + 12\).
60 Example - Factoring a x^2 + bx + c by Trial and Error · Level 3
Factor: \(6x^2 + 7x - 5\).
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61 Example - Recognizing the Form of an Expression · Level 3
Factor each expression. (a) \(x^2 - 2x - 3\) (b) \((5a + 1)^2 - 2(5a + 1) - 3\)
62 Example - Factoring Differences of Squares · Level 2
Factor each expression. (a) \(4x^2 - 25\) (b) \((x + y)^2 - z^2\)
63 Example - Factoring Differences and Sums of Cubes · Level 3
Factor each polynomial. (a) \(27x^3 - 1\) (b) \(x^6 + 8\)
64 Example - Recognizing Perfect Squares · Level 2
Factor each trinomial. (a) \(x^2 + 6x + 9\) (b) \(4x^2 - 4 x y + y^2\)

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