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As you have previous learned, a monomial is one number that stands on its own and a polynomial is two or more monomial numbers that are linked together in an equation by an addition sign (+), subtraction sign (-) or a multiplication sign (x). For example: 3*x*^{2} ● 5*x*^{5}. The “●” symbol will be used here in place of the normal “x” used for multiplication so as to not confuse “*x*” with “x” in algebra problems.

When multiplying monomials, any exponents are added together. So in the example 3*x*^{2} ● 5*x*^{5}, “^{2}” and “^{5}” will be added and the equations will then look like:

3*x*^{2} ● 5*x*^{5}

3 ● 5 = 15*x*^{7}

When multiplying polynomials, any exponents are added. Let’s look at the following example: (3*x*^{3})(4*x*^{4}). [**Note:** This is a polynomial. When the “●” is replaced with parentheses(), the numbered problem becomes a polynomial and the parentheses indicate that we will be multiplying the two monomial numbers.] This problem is worked as follows to find the solution.

(3*x*^{3})(4*x*^{4})

3 ● 4 = 12

*x*^{3} + *x*^{4} = *x*^{7}

** Solution:** 12

Now let’s look at the next example of 2*x*(3*x* + 7) and see how to find the solution.

2*x*(3*x* + 7)

(2*x*)(3*x*) + (2*x*)(7)

6*x*^{2} + 14*x*

1.

7*x*(4*x*^{2} + 3*x* + 12)

28*x*^{3} + 21*x*^{2} - 84*x*

28*x*^{3} + 21*x* + 84*x*

11*x*^{3} + 10*x*^{2} + 19*x*

28*x*^{3} + 21*x*^{2} + 84*x*

2.

(8*x*^{5})(3*x*^{2})

24*x*^{10}

11*x*^{7}

24*x*^{7}

11*x*^{10}

(8*x*^{5})(3*x*^{2})

8 ● 3 = 24

*x*^{5} + *x*^{2} = *x*^{7}

__Solution__: 24*x*^{7}

Answer (c) is the correct solution

8 ● 3 = 24

Answer (c) is the correct solution

3.

15*x*^{2} ● 5*x*^{8}

75*x*^{10}

20*x*^{10}

20*x*^{16}

75*x*^{16}

15*x*^{2} ● 5*x*^{8}

15 ● 5 = 75*x*^{10} (remember to add exponents)

__Solution__: 75*x*^{10}

Answer (a) is the correct solution

15 ● 5 = 75

Answer (a) is the correct solution

4.

3*x*(17*x*^{4} + 5*x*^{2} + 13)

51*x*^{5} + 15*x* + 39*x*^{2}

51*x*^{8} + 15*x*^{2} + 39*x*^{2}

51*x*^{4} + 15*x*^{2} + 39*x*

51*x*^{5} + 15*x*^{3} + 39*x*

3*x*(17*x*^{4} + 5*x*^{2} + 13)

(3*x*)(17*x*^{4}) = 51*x*^{5} (add exponents)

(3*x*)(5*x*^{2}) = 15*x*^{3} (add exponents)

(3*x*)(13) = 39*x*

__Solution__: 51*x*^{5} + 15*x*^{3} + 39*x*

Answer (d) is the correct solution

(3

(3

(3

Answer (d) is the correct solution

5.

20*x*^{3} - 8*x*^{2} + 12*x*

20*x*^{2} - 8*x* + 12*x*

9*x*^{3} - 2*x*^{2} + 7*x*

9*x*^{3} - 8*x*^{2} + 12*x*

4*x*(5*x*^{2} - 2*x* + 3)

(4*x*)(5*x*^{2}) = 20*x*^{3}(add exponents)

(4*x*)(-2*x*) = -8*x*^{2} (add exponents)

(4*x*)(3) = 12*x*

__Solution__: 20*x*^{3} - 8*x*^{2} + 12*x*

Answer (a) is the correct solution

(4

(4

(4

Answer (a) is the correct solution

6.

6*x*^{5} ● 3*x*^{4}

18*x*^{20}

18*x*^{9}

9*x*^{9}

9*x*^{20}

6*x*^{5} ● 3*x*^{4}

6 ● 3 = 18*x*^{9} (remember to add exponents)

__Solution__: 18*x*^{9}

Answer (b) is the correct solution

6 ● 3 = 18

Answer (b) is the correct solution

7.

(21*x*^{9})(8*x*^{3})

168*x*^{27}

168*x*^{12}

168*x*^{6}

168*x*^{9}

(21*x*^{9})(8*x*^{3})

21 ● 8 = 168

*x*^{9} + *x*^{3} = *x*^{12}

__Solution__: 168*x*^{12}

Answer (b) is the correct solution

21 ● 8 = 168

Answer (b) is the correct solution

8.

20*x*^{3} + 10*x* + 100*x*^{2}

20*x*^{2} + 10*x* + 100*x*

20*x*^{3} + 10*x*^{2} + 100*x*

20*x*^{2} + 10*x*^{2} + 100*x*

(

(

(

Answer (c) is the correct solution

9.

10*x*(5*x*^{2} - 9*x* + 22)

50*x*^{3} + 90*x*^{2} + 220*x*

50*x*^{3} - 90*x* + 220*x*

50*x*^{3} - 90*x*^{2} + 220*x*

50*x*^{3} - 90*x*^{2} + 120*x*

10*x*(5*x*^{2} - 9*x* + 22)

(10*x*)(5*x*^{2}) = 50*x*^{3} (add exponents)

(10*x*)(-9*x*) = -90*x*^{2} (add exponents)

(10*x*)(22) = 220*x*

__Solution__: 50*x*^{3} - 90*x*^{2} + 220*x*

Answer (c) is the correct solution

(10

(10

(10

Answer (c) is the correct solution

10.

12*x*^{4} ● 10*x*^{2}

120*x*^{6}

102*x*^{8}

120*x*^{8}

22*x*^{6}

12*x*^{4} ● 10*x*^{2}

12 ● 10 = 120*x*^{6} (remember to add exponents)

__Solution__: 120*x*^{6}

Answer (a) is the correct solution

12 ● 10 = 120

Answer (a) is the correct solution

The next step is:

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x(4x^{2}+ 3x+ 12)(7

x)(4x^{2}) = 28x^{3}(add exponents)(7

x)(3x) = 21x^{2}(add exponents)(7

x)(12) = 84xSolution: 28x^{3}+ 21x^{2}+ 84xAnswer (d) is the correct solution