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8.2: 多项式乘法

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    171122

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    多项式可以分类为：

    • 单项式（如果它们包含一个术语）。
    • 二项式，如果它们包含两个项。
    • 如果三项式包含三个项。
    • 多项式，如果它们包含三个或更多项。

    本节中没有示例或作业。

    两个单项式的乘法

    定义：将两个单项式相乘

    要将两个单项式相乘，请通过将指数相加并乘以数值系数将两项相乘。

    示例 Template:Index

    将两个单项式相乘：

    1. \((3x^2 )(6x^3 )\)
    2. \((4x)(x)\)
    3. \((−2x^3 )(−7x^4 )\)
    解决方案
    1. \(\begin{array} &&(3x^2 )(6x^3 ) &\;\;\;\;\;\;\;\;\;\;\;\;\text{Example problem} \\ &(3)(6)(x^{2+3}) &\;\;\;\;\;\;\;\;\;\;\;\;\text{Multiply the coefficients and add the exponents on the variables using the Product Rule for Exponents} \\ &18x^5 &\;\;\;\;\;\;\;\;\;\;\;\;\text{Solution} \end{array}\)
    1. \(\begin{array} &&(4x)(x) &\;\;\;\;\;\;\;\;\;\;\;\;\text{Example problem} \\ &(4)(1)(x^{1+1}) &\;\;\;\;\;\;\;\;\;\;\;\;\text{Multiply the coefficients and add the exponents. The coefficient on \(x\)是\(1\)，每一个的指数\(x\)是\(1\)。}\\ &4x^2 &\;\;\;\;\;\;\;\;\;\;\;\ text {array}\)
    1. \(\begin{array} &&(−2x^3 )(−7x^4 ) &\;\;\;\;\;\;\;\;\text{Example problem} \\ &(−2)(−7)(x^{3+4}) &\;\;\;\;\;\;\;\;\text{Multiply the coefficients and add the exponents.} \\ &14x^7 &\;\;\;\;\;\;\;\;\text{Solution} \end{array}\)
    练习 Template:Index

    将两个单项式相乘：

    1. \((−3x^4 )(9x^7 )\)
    2. \((2x)(2x)\)
    3. \((−4x^7 )(5x^5 )\)
    4. \((−6x^2 )(−x^2 )\)

    将多项式乘以单项式

    定义：将多项式乘以单项式

    要将多项式乘以单项式，请将多项式的所有项乘以单项式。 保留原始多项式中的所有减法，减法后的项作为该项系数的符号。

    示例 Template:Index

    将多项式乘以单项式：

    1. \(3x^2 (15x^2 − 5x)\)
    2. \(−7x(3x^2 − 2x + 9)\)
    3. \(5x(4x^3 − 2x^2 + x − 3)\)
    解决方案
    1. \(\begin{array} &&3x^2 (15x^2 − 5x) &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Example problem} \\ &(3x^2 )(15x^2 ) + (3x^2 )(−5x) &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Multiply all terms of the polynomial by the monomial. Then simplify by multiplying the pairs of monomials.} \\ &45x^4 + (−15x^3 ) &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Simplify} \\ &45x^4 − 15x^3 &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Solution} \end{array}\)
    1. \(\begin{array} &&−7x(3x^2 − 2x + 9) &\;\;\;\;\;\;\;\;\;\;\text{Example problem} \\ &(−7x)(3x^2 ) + (−7x)(−2x) + (−7x)(9) &\;\;\;\;\;\;\;\;\;\;\text{Multiply the coefficients and add the exponents. The coefficient on \(x\)是\(1\)，每个指数的指数\(x\)是\(1\)。}\\ &−21x^3 + 14x^2 − 63x &\;\;\;\;\;\;\;\;\;\;\;\;\ text {Solution}\ end {array}\)
    1. \(\begin{array} &&5x(4x^3 − 2x^2 + x − 3) &\text{Example problem} \\ &(5x)(4x^3 ) + (5x)(−2x^2 ) + (5x)(x) + (5x)(−3) &\text{Multiply the coefficients and add the exponents.} \\ &20x^4 − 10x^3 + 5x^2 − 15x &\text{Solution} \end{array}\)
    练习 Template:Index

    将多项式乘以单项式：

    1. \((−6x)(x^2 − 3)\)
    2. \((3x^4 )(2x^2 − x − 5)\)
    3. \((−4x^5 )(x^4 − 3x^3 + 3x^2 − x − 7)\)
    4. \((x^2 )(−x^3 − 12)\)

    两个二项式的乘法

    定义：将两个二项式相乘

    要将两个二项式相乘，请使用 FO IL 技术进行乘法：第一个项、外部项、内部项和最后一个项。 FOIL 确保将第一个二项式中的所有项与第二个二项式中的所有项相乘。 项的乘法顺序并不重要，因为乘法是可交换的。 注意将任何类似的术语组合在一起，以完全简化解决方案。

    示例 Template:Index

    将两个二项式相乘：

    1. \((3x − 4)(2x + 5)\)
    2. \((5x^2 − 2)(5x^2 + 2)\)
    3. \((7x^3 − 4x^2 )(x − 5)\)
    解决方案
    1. \(\begin{array} &&(3x − 4)(2x + 5) &\;\;\;\;\;\;\;\;\;\;\text{Example problem} \\ &(3x)(2x) + (3x)(5) + (−4)(2x) + (−4)(5) &\;\;\;\;\;\;\;\;\;\;\text{FOIL the terms to multiply all terms in the first binomial by all terms in the second binomial.} \\ &6x^2 + 15x + (−8x) + (−20) &\;\;\;\;\;\;\;\;\;\;\text{Combine like terms and simplify} \\ &6x^2 + 7x − 20 &\;\;\;\;\;\;\;\;\;\;\text{Solution} \end{array}\)
    1. \(\begin{array} &&(5x^2 − 2)(5x^2 + 2) &\;\;\;\;\text{Example problem} \\ &(5x^2 )(5x^2 ) + (5x^2 )(2) + (−2)(5x^2 ) + (−2)(2) &\;\;\;\;\text{FOIL the terms to multiply all terms in the first binomial by all terms in the second binomial.} \\ &25x^4 + 10x^2 + (−10x^2 ) + (−4) &\;\;\;\;\text{Combine like terms and simplify} \\ &25x^4 − 4 &\;\;\;\;\text{Solution} \end{array}\)
    1. \(\begin{array} &&(7x^3 − 4x^2 )(x − 5) &\text{Example problem} \\ &(7x^3 )(x) + (7x^3 )(−5) + (−4x^2 )(x) + (−4x^2 )(−5) &\text{FOIL the terms to multiply all terms in the first binomial by all terms in the second binomial.} \\ &7x^4 + (−35x^3 ) + (−4x^3 ) + 20x^2 &\text{Combine like terms and simplify} \\ &7x^4 − 39x^3 + 20x^2 &\text{Solution} \end{array}\)
    练习 Template:Index

    将两个二项式相乘：

    1. \((2x − 3)(6x + 5)\)
    2. \((3x^2 − 4)(3x^2 + 4)\)
    3. \((−4x^5 − 2)(7x^3 + 3)\)
    4. \((2x − 7)(3x − 8)\)

    两个多项式的乘法

    定义：将两个多项式相乘

    要将两个多项式相乘，请使用分布属性将第一个多项式中的每个项乘以第二个多项式中的每个项。 然后将相似的术语组合在一起以简化解决方案。

    示例 Template:Index

    将两个多项式相乘：

    1. \((2x + 5)(3x^2 − 6x + 9)\)
    2. \((2x^2 + 4x − 5)(3x − 2)\)
    3. \((x^2 − x + 3)(2x^2 + 6x − 1)\)
    解决方案
    1. \(\begin{array} &&(2x + 5)(3x^2 − 6x + 9) &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Example problem} \\ &(2x)(3x^2 ) + (2x)(−6x) + (2x)(9) + (5)(3x^2 ) + (5)(−6x) + (5)(9) &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{FOIL the terms to multiply all terms in the} \\ & &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \text{first binomial by all terms in the second binomial.} \\ &6x^3 + (−12x^2 ) + 18x + 15x^2 + (−30x) + 45 &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Combine like terms and simplify} \\ &6x^3 + 3x^2 − 12x + 45 &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Solution} \end{array}\)
    1. \(\begin{array} &&(2x^2 + 4x − 5)(3x − 2) &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Example problem} \\ &(2x^2 )(3x) + (2x^2 )(−2) + (4x)(3x) + (4x)(−2) + (−5)(3x) + (−5)(−2) &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{FOIL the terms to multiply all terms in the} \\ & &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \text{first binomial by all terms in the second binomial.} \\ &6x^3 + (−4x^2 ) + 12x^2 + (−8x) + (−15x) + 10 &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Combine like terms and simplify} \\ &6x^3 + 8x^2 − 23x + 10 &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Solution} \end{array}\)
    1. \(\begin{array} &&(x^2 − x + 3)(2x^2 + 6x − 1) &\text{Example problem} \\ &(x^2 )(2x^2 ) + (x^2 )(6x) + (x^2 )(−1) + (−x)(2x^2 ) + (−x)(6x) + (−x)(−1) + (3)(2x^2 ) + (3)(6x) + (3)(−1) &\text{FOIL the terms to multiply all terms in the} \\ & & \text{first binomial by all terms in the second binomial.} \\ &2x^4 + 6x^3 + (−1x^2 ) + (−2x^3 ) + (−6x^2 ) + x + 6x^2 + 18x + (−3) &\text{Combine like terms and simplify} \\ &2x^4 + 4x^3 − x^2 + 19x − 3 &\text{Solution} \end{array}\)
    练习 Template:Index

    将两个多项式相乘：

    1. \((x^2 − 2x − 1)(2x^2 − 7x − 8)\)
    2. \((3x^2 − 5)(x^2 + 4x − 3)\)
    3. \((4x^3 − 2x + 1)(6x^2 + 3)\)
    4. \((2x^3 − 3x + 4)(2x^2 − 8x + 2)\)
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    This page titled 8.2: 多项式乘法 is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Victoria Dominguez, Cristian Martinez, & Sanaa Saykali (ASCCC Open Educational Resources Initiative) .