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Mathematics LibreTexts

7.5E: Exercises

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    30445
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    Practice Makes Perfect

    Recognize and Use the Appropriate Method to Factor a Polynomial Completely

    In the following exercises, factor completely.

    Exercise \(\PageIndex{31}\)

    \(10x^4+35x^3\)

    Answer

    \(5x^{3}(2x+7)\)

    Exercise \(\PageIndex{32}\)

    \(18p^6+24p^3\)

    Exercise \(\PageIndex{33}\)

    \(y^2+10y−39\)

    Answer

    (y−3)(y+13)

    Exercise \(\PageIndex{34}\)

    \(b^2−17b+60\)

    Exercise \(\PageIndex{35}\)

    \(2n^2+13n−7\)

    Answer

    (2n−1)(n+7)

    Exercise \(\PageIndex{36}\)

    \(8x^2−9x−3\)

    Exercise \(\PageIndex{37}\)

    \(a^5+9a^3\)

    Answer

    \(a^{3}(a^2+9)\)

    Exercise \(\PageIndex{38}\)

    \(75m^3+12m\)

    Exercise \(\PageIndex{39}\)

    \(121r^2−s^2\)

    Answer

    (11r−s)(11r+s)

    Exercise \(\PageIndex{40}\)

    \(49b^2−36a^2\)

    Exercise \(\PageIndex{41}\)

    \(8m^2−32\)

    Answer

    8(m−2)(m+2)

    Exercise \(\PageIndex{42}\)

    \(36q^2−100\)

    Exercise \(\PageIndex{43}\)

    \(25w^2−60w+36\)

    Answer

    \((5w−6)^2\)

    Exercise \(\PageIndex{44}\)

    \(49b^2−112b+64\)

    Exercise \(\PageIndex{45}\)

    \(m^2+14mn+49n^2\)

    Answer

    \((m+7n)^2\)

    Exercise \(\PageIndex{46}\)

    \(64x^2+16xy+y^2\)

    Exercise \(\PageIndex{47}\)

    \(7b^2+7b−42\)

    Answer

    7(b+3)(b−2)

    Exercise \(\PageIndex{48}\)

    \(3n^2+30n+72\)

    Exercise \(\PageIndex{49}\)

    \(3x^3−81\)

    Answer

    \(3(x−3)(x^2+3x+9)\)

    Exercise \(\PageIndex{50}\)

    \(5t^3−40\)

    Exercise \(\PageIndex{51}\)

    \(k^4−16\)

    Answer

    \((k−2)(k+2)(k^2+4)\)

    Exercise \(\PageIndex{52}\)

    \(m^4−81\)

    Exercise \(\PageIndex{53}\)

    \(15pq−15p+12q−12\)

    Answer

    3(5p+4)(q−1)

    Exercise \(\PageIndex{54}\)

    \(12ab−6a+10b−5\)

    Exercise \(\PageIndex{55}\)

    \(4x^2+40x+84\)

    Answer

    4(x+3)(x+7)

    Exercise \(\PageIndex{56}\)

    \(5q^2−15q−90\)

    Exercise \(\PageIndex{57}\)

    \(u^5+u^2\)

    Answer

    \(u^{2}(u+1)(u^2−u+1)\)

    Exercise \(\PageIndex{58}\)

    \(5n^3+320\)

    Exercise \(\PageIndex{59}\)

    \(4c^2+20cd+81d^2\)

    Answer

    prime

    Exercise \(\PageIndex{60}\)

    \(25x^2+35xy+49y^2\)

    Exercise \(\PageIndex{61}\)

    \(10m^4−6250\)

    Answer

    \(10(m−5)(m+5)(m^2+25)\)

    Exercise \(\PageIndex{62}\)

    \(3v^4−768\)

    Everyday Math

    Exercise \(\PageIndex{63}\)

    Watermelon drop A springtime tradition at the University of California San Diego is the Watermelon Drop, where a watermelon is dropped from the seventh story of Urey Hall.

    1. The binomial \(−16t^2+80\) gives the height of the watermelon t seconds after it is dropped. Factor the greatest common factor from this binomial.
    2. If the watermelon is thrown down with initial velocity 8 feet per second, its height after t seconds is given by the trinomial \(−16t2−8t+80\)
    Answer
    1. \(−16(t^2−5)\)
    2. −8(2t+5)(t−2)

    Exercise \(\PageIndex{64}\)

    Pumpkin drop A fall tradition at the University of California San Diego is the Pumpkin Drop, where a pumpkin is dropped from the eleventh story of Tioga Hall.

    1. The binomial \(−16t^2+128\) gives the height of the pumpkin t seconds after it is dropped. Factor the greatest common factor from this binomial.
    2. If the pumpkin is thrown down with initial velocity 32 feet per second, its height after t seconds is given by the trinomial \(−16t^2−32t+128\)

    Writing Exercises

    Exercise \(\PageIndex{65}\)

    The difference of squares \(y^4−625\) can be factored as \((y^2−25)(y^2+25)\) completely factored. What more must be done to completely factor it?

    Exercise \(\PageIndex{66}\)

    Of all the factoring methods covered in this chapter (GCF, grouping, undo FOIL, ‘ac’ method, special products) which is the easiest for you? Which is the hardest? Explain your answers.

    Self Check

    a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This table has the following statements all to be preceded by “I can…”. The row states “recognize and use the appropriate method to factor a polynomial completely”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.

    b. Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

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