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Integral expressions (polynomials)
The degree of a monomial is the sum of the indices of the variable(s). The degree of a polynomial is the degree of the term with the largest degree.
For example
- expression \latex{3x^{5} -6x+13 } has a degree of five,
- expression \latex{-2a+3a^{2} -2a^{7}-6} has a degree of seven because of the third term,
- expression \latex{4x^{2}yz^{2}+2xyz-3y^{3}x} has a degree of five because of the first term,
- expression \latex{-abc+2a^{2}-b-3cb} has a degree of three because of the first term.
Two monomials are like terms if at most their coefficients differ. In a polynomial we can combine the like terms.
Example 1
Let us combine like terms in the following expressions.
- \latex{-4x^{3}+2x^{2}-5x+3x^{3}-4x+2x^{2}}
- \latex{\left(-2x^{3}+3x^{2}-5x+7\right)+\left(5x^{3}-4x^{2}+x-2\right)}
- \latex{\left(-2x^{3}y+3x^{2}y^{2}-5x+7y \right)-\left(5x^{3}y-4x^{2}y^{2}+x-2y\right)}
Solution
- \latex{-4x^{3}+2x^{2}-5x+3x^{3}-4x+2x^{2}=-4x^{3}+3x^{3}+2x^{2}+2x^{2}-5x-4x=}\latex{=\left(-4+3\right)x^{3}+\left(2+2\right)x^{2}+\left(-5-4\right)x=-x^{3}+4x^{2}-9x}
- \latex{-2x^{3}+3x^{2}-5x+7+5x^{3}-4x^{2}+x-2=}\latex{=-2x^{3}+3x^{2}-5x+7+5x^{3}-4x^{2}+x-2=}\latex{=\left(-2+5\right)x^{3}+\left(3-4\right)x^{2}+\left(-5+1\right)x+5=3x^{3} -x^{2} -4x+5}
- \latex{-2x^{3}y+3x^{2}y^{2}-5x+7y-5x^{3}y-4x^{2}y^{2}+x-2y=}\latex{=-2x^{3}y+3x^{2}y^{2}-5x+7y-5x^{3}y+4x^{2}y^{2}-x+2y=}\latex{=\left(-2-5\right)x^{3}y+\left(3+4\right)x^{2}y^{2}+\left(-5-1\right)x+\left(7+2\right)y=} \latex{=-7x^{3}y+7x^{2}y^{2}-6x+9y}
When multiplying two integral expressions we can use that multiplication is distributive over addition, i.e. we can multiply the sum by terms.
Example 2
Let us do the following multiplications.
- \latex{2a\left(3a^{2}+1\right)}
- \latex{\left(4a-3\right)\times a^{2}}
- \latex{\left(4x-3\right)\left(2x^{2}-3x+1\right)}
Solution
a) \latex{2a\left(3a^{2}+1 \right)=2a\times 3a^{2}+2a\times 1=6a^{3}+2a};
b) \latex{\left(4a-3\right)\times a^{2}=4a\times a^{2}-3\times a^{2}=4a^{3}-3a^{2}};
c) \latex{\left(4x-3\right)\left(2x^{2}-3x+1 \right)=4x\times 2x^{2}+4x\times \left(-3x\right)+4x\times 1+}
\latex{+\left(-3\right)\times 2x^{2}+\left(-3\right)\times \left(-3x\right)+\left(-3\right)\times 1=}
\latex{=8x^{3}-12x^{2}+4x-6x^{2}+9x-3=8x^{3}-18x^{2}+13x-3}.
\latex{+\left(-3\right)\times 2x^{2}+\left(-3\right)\times \left(-3x\right)+\left(-3\right)\times 1=}
\latex{=8x^{3}-12x^{2}+4x-6x^{2}+9x-3=8x^{3}-18x^{2}+13x-3}.
Here we do not deal with the division of two integral expressions, since it gives an algebraic fractional expression and not an integral expression.
Exercises
{{exercise_number}}. Put the below polynomials into ascending order based on their degrees.
\latex{-2d^{3}+3}
\latex{11x^{4}y^{2}}
\latex{0.4a^{2}-2b}
\latex{2.3g^{2}-3g^{4}}
\latex{38s^{3}t^{2}-7s^{2}t}
{{exercise_number}}. In the below expressions combine the like terms and then put the terms into descending order based on their degree.
- \latex{2y-3y^{2}+5y-6+y^{2}+3-3y+4y^{2}}
- \latex{3x^{2}+2x^{3}-5x+4x^{3}-5x^{2}-3+x^{2}-x^{3}+4x-1}
- \latex{2a^{2}b-3ab-4b^{2}+3ba^{2}+5b^{2}-4a^{2}b+4ab+a^{2}b-2ab}
{{exercise_number}}. Do the following operations, combine the like terms in the results, and put the terms into descending order based on their degrees.
- \latex{\left(2y-3\right)+\left(3-4y\right)-\left(y+1\right)}
- \latex{\left(4x-3x^{2}+5\right)-\left(4-2x+2x^{2}\right)-\left(x^{2}-3x+1\right)}
- \latex{\left(3ab-4ab^{2}+2a^{2}b\right)+\left(3a^{2}b-2ab-5ab^{2}\right)-\left(2ab^{2}-5ab\right)}
- \latex{\left(2-a^{2}+5a\right)-\left(3a-a^{2}+7\right)-\left(a^{2}+3a-5\right)}
- \latex{\left(x^{2}-3x^{3}+8x\right)+\left(x^{3}-7x+3\right)-\left(x^{2}-6x+1\right)-\left(2-7x+x^{3} \right)}
- \latex{\left(xy^{2}-3xy+7x^{2}y\right)-\left(2yx^{2}-3xy-2y^{2}x\right)+\left(7xy-2x^{2}y+y^{2}x\right)}
{{exercise_number}}. Do the following operations, combine the like terms in the results, and put the terms into descending order based on their degrees.
- \latex{4a\left(7a-3\right)}
- \latex{3x\left(2x^{2}-3x+7\right)}
c) \latex{\left(2a^{2}-7+a\right)\times3a}
- \latex{\left(3a-2\right)\left(3-2a\right)}
- \latex{\left(2x+3\right)\left(3x^{2}-6x+5\right)}
- \latex{\left(4x^{2}+x-2\right)\left(2x^{2}+3x+1\right)}
{{exercise_number}}. Do the following operations, combine the like terms in the results, and put the terms into descending order based on their degrees.
- \latex{\left(a+2\right)\left(a+2\right)}
- \latex{\left(7a-3\right)\left(7a-3\right)}
- \latex{\left(8a+1\right)\left(8a-1\right)}
- \latex{\left(a+b+1\right)\left(a+b+1\right)}
{{exercise_number}}. What is the sum of the coefficients in the polynomial form of the expression \latex{\left(x^{2}-2x+2\right)^{2000}?}