# 8.2: Kuzidisha Polynomial

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Polynomials inaweza kuwa classified kama:

• Monomials ikiwa yana muda mmoja.
• Binomials kama yana mbili mrefu.
• Trinomials kama yana tatu mrefu.
• Polynomials ikiwa yana maneno matatu au zaidi.

Hakuna mifano au kazi za nyumbani katika sehemu hii.

## Kuzidisha kwa Monomials mbili

##### Ufafanuzi: Kuzidisha Monomials mbili

Ili kuzidisha monomials mbili, kuzidisha maneno pamoja kwa kuongeza exponents na kuzidisha coefficients numeric.

##### Mfano Template:index

Panua monomials mbili:

1. $$(3x^2 )(6x^3 )$$
2. $$(4x)(x)$$
3. $$(−2x^3 )(−7x^4 )$$
###### Suluhisho
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$$ni$$1$$, na exponent juu ya kila$$x$$ ni$$1$$.}\\ &4x^2 &\;\;\;\;\;\;\;\;\;\;\;\ maandishi {Solution}\ mwisho {safu}\)
1. $$\begin{array} &&(−2x^3 )(−7x^4 ) &\;\;\;\;\;\;\;\;\text{Example problem} \\ &(−2)(−7)(x^{3+4}) &\;\;\;\;\;\;\;\;\text{Multiply the coefficients and add the exponents.} \\ &14x^7 &\;\;\;\;\;\;\;\;\text{Solution} \end{array}$$
##### Zoezi Template:index

Panua monomials mbili:

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

## Kuzidisha kwa Polynomial na Monomial

##### Ufafanuzi: Kuzidisha Polynomial Kwa Monomial

Ili kuzidisha polynomial na monomial, kuzidisha masharti yote ya polynomial na monomial. Weka vikwazo vyovyote katika polynomial ya awali na neno linalofuata uondoaji kama ishara ya mgawo wa muda.

##### Mfano Template:index

Panua polynomial na monomial:

1. $$3x^2 (15x^2 − 5x)$$
2. $$−7x(3x^2 − 2x + 9)$$
3. $$5x(4x^3 − 2x^2 + x − 3)$$
###### Suluhisho
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$$ni$$1$$, na exponent juu ya kila$$x$$ ni$$1$$.}\\ &-21x ^ 3 + 14x ^ 2 - 63x &\;\;\;\;\;\;\;\;\;\ Nakala {Solution}\ mwisho {safu}\)
1. $$\begin{array} &&5x(4x^3 − 2x^2 + x − 3) &\text{Example problem} \\ &(5x)(4x^3 ) + (5x)(−2x^2 ) + (5x)(x) + (5x)(−3) &\text{Multiply the coefficients and add the exponents.} \\ &20x^4 − 10x^3 + 5x^2 − 15x &\text{Solution} \end{array}$$
##### Zoezi Template:index

Panua polynomial na monomial:

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

## Kuzidisha kwa Binomials mbili

##### Ufafanuzi: Kuzidisha Binomials mbili

Ili kuzidisha binomials mbili, tumia mbinu ya FOIL ili kuzidisha: maneno ya kwanza, maneno ya nje, maneno ya ndani na masharti ya mwisho. FOIL inahakikisha kwamba maneno yote katika binomial ya kwanza yanaongezeka kwa maneno yote katika binomial ya pili. Utaratibu wa kuzidisha maneno haijalishi tangu kuzidisha ni kubadilisha. Jihadharini kuchanganya maneno kama hayo ili kurahisisha kikamilifu suluhisho.

##### Mfano Template:index

Panua binomials mbili:

1. $$(3x − 4)(2x + 5)$$
2. $$(5x^2 − 2)(5x^2 + 2)$$
3. $$(7x^3 − 4x^2 )(x − 5)$$
###### Suluhisho
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. $$\begin{array} &&(7x^3 − 4x^2 )(x − 5) &\text{Example problem} \\ &(7x^3 )(x) + (7x^3 )(−5) + (−4x^2 )(x) + (−4x^2 )(−5) &\text{FOIL the terms to multiply all terms in the first binomial by all terms in the second binomial.} \\ &7x^4 + (−35x^3 ) + (−4x^3 ) + 20x^2 &\text{Combine like terms and simplify} \\ &7x^4 − 39x^3 + 20x^2 &\text{Solution} \end{array}$$
##### Zoezi Template:index

Panua binomials mbili:

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

## Kuzidisha kwa Polynomials mbili

##### Ufafanuzi: Kuzidisha Polynomials mbili

Ili kuzidisha polynomials mbili, tumia mali ya usambazaji ili kuzidisha kila neno katika polynomial ya kwanza kwa kila neno katika polynomial ya pili. Kama maneno ni kisha pamoja ili kurahisisha ufumbuzi.

##### Mfano Template:index

Panua polynomials mbili:

1. $$(2x + 5)(3x^2 − 6x + 9)$$
2. $$(2x^2 + 4x − 5)(3x − 2)$$
3. $$(x^2 − x + 3)(2x^2 + 6x − 1)$$
###### Suluhisho
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. $$\begin{array} &&(x^2 − x + 3)(2x^2 + 6x − 1) &\text{Example problem} \\ &(x^2 )(2x^2 ) + (x^2 )(6x) + (x^2 )(−1) + (−x)(2x^2 ) + (−x)(6x) + (−x)(−1) + (3)(2x^2 ) + (3)(6x) + (3)(−1) &\text{FOIL the terms to multiply all terms in the} \\ & & \text{first binomial by all terms in the second binomial.} \\ &2x^4 + 6x^3 + (−1x^2 ) + (−2x^3 ) + (−6x^2 ) + x + 6x^2 + 18x + (−3) &\text{Combine like terms and simplify} \\ &2x^4 + 4x^3 − x^2 + 19x − 3 &\text{Solution} \end{array}$$
##### Zoezi Template:index

Panua polynomials mbili:

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. $$(2x^3 − 3x + 4)(2x^2 − 8x + 2)$$