Multiplication of Polynomial by Monomial
Rule for Multiplication of Polynomial by Monomial
To multiply a polynomial by a monomial , multiply each term of the polynomial by the monomial and sum up the results.
By this rule, you can also multiply a monomial by a polynomial, since the factors can be interchanged.
How to multiply a monomial by another monomial (a term of a polynomial) – see §13 of this Guide.
Examples of Problem Solving
Example 1. Find the product of the polynomial and the monomial::
a) $y^3-5y^2+4 and 2y$
$(y^3-5y^2+4 ) \cdot 2y = y^3 \cdot 2y-5y^2 \cdot 2y+4 \cdot 2y = 2y^4-10y^3+8y $
b) $ a^2+ab+b^2 and 2ab $
$ (a^2+ab+b^2 ) \cdot 2ab = a^2 \cdot 2ab+ab \cdot 2ab+b^2 \cdot 2ab = 2a^3 b+2a^2 b^2+2ab^3 $
c) $\frac{1}{3} x^4+5x^3-4 and \frac{1}{4} x^2 y$
$ (\frac{1}{3} x^4+5x^3-4 )\cdot \frac{1}{4} x^2 y = \frac{1}{3} x^4 \cdot \frac{1}{4} x^2 y+5x^3 \cdot \frac{1}{4} x^2 y-4\cdot \frac{1}{4} x^2 y = \frac{1}{12} x^6 y+1 \frac{1}{4} x^5 y-x^2 y $
d) $ 0,8t^4+1,5t-0,2 and 10t^3$
$ (0,8t^4+1,5t-0,2 )\cdot 10t^3 = 0,8t^4 \cdot 10t^3+1,5t \cdot 10t^3-0,2 \cdot 10t^3 = 8t^7+15t^4-2t^3 $
Example 2. Simplify the expression:
a) $ 3xy+2x^2 y-xy(3+2x)+7 = 3xy+2x^2 y-3xy-2x^2 y+7 = 7 $
b) $ x^3-x(x^2-1)-0,5 = x^3-x^3+x-0,5 = x-0,5 $
c) $ 3ax^2+ax-ax(3x-5) = 3ax^2+ax-3ax^2+5ax = 6ax $
d) $ \frac{1}{5} x^2 y+x^3-x^2 (y+x) = \frac{1}{5} x^2 y+x^3-x^2 y-x^3 = -\frac{4}{5} x^2 y $
Example 3. Solve the equation:
a) 2(x+8)-3(x+5) = x-1
2x+16-3x-15 = x-1
2x-3x-x = -1-16+15
-2x = -2
x = 1
b) $ \frac{1}{2} (x+5)-4(8-x) = 12,5+x $
$ \frac{1}{2} x+\frac{5}{2}-32+4x = 125+x $
0,5x+4x-x = 12,5-2,5+32
3,5x = 42
x = 12
Example 4. Prove that the value of the expression does not depend on the variable:
a) $3(x^3+4)-5x(x^2+x)+2x^2 (x+2,5) = 3x^3+12-5x^3-5x^2+2x^3+5x^2 = 12$
b) $5ab-4(a^2-1)+a(4a-5b) = 5ab-4a^2+4+4a^2-5ab = 4$
Example 5. The sum of two numbers is 50. If the first number is multiplied by 7 and the second by 2, the sum of these products is 195. Find the given numbers
Solution:
Let x be the first number. Then, the second is 50-x. Solve the equation:
7x+2(50-x) = 195
7x+100-2x = 195
5x = 195-100
5x = 95
x = 19 - the first number
50-x = 50-19 = 31 - the second number
Answer: 19 and 31
Example 6. How old is a man if, in 20 years, he will be 3 times older than he was 20 years ago?
Solution:
Let x be the current age. As stated:
x+20 = 3(x-20)
x+20 = 3x-60
x-3x = -60-20
-2x = -80
x = 40
Answer: 40 years old
Example 7. Prove that for any value of the variable, the expression is always positive:
$$ 5x(3x^2-1)-6x(x^2+1)-3x(3x^2+7)+32(x+1) = $$
$$ = 15x^3-5x-6x^3-6x-9x^3-21x+32x+32 = 32 \gt 0 $$
The value of the expression does not depend on the variable; it is constant and positive.
Example 8*. Find the value of the parameter a such that the polynomial $2y^3+5ay^2-ay-1$ has the same numerical values for y=1 and y=-1?
Denote by P(y;a) = $2y^3+5ay^2-ay-1$
Substitute the given values of y.
$ P(1;a) = 2\cdot1^3+5a\cdot1^2-a\cdot1-1 = 4a+1 $
$P(-1;a) = 2 \cdot (-1)^3+5a \cdot (-1)^2-a \cdot (-1)-1 = 6a-3$
As stated: $ P(1;a) = P(1;a) \Rightarrow 4a+1 = 6a-3 \Rightarrow 2a = 4 \Rightarrow a = 2 $
Answer: 2
