Division Of Polynomials

Division of Polynomials,[object Object],Division of polynomials may look difficult, but it follows the same rules we have already learned for properties of real numbers as well as for division of real numbers.,[object Object]

Divide a Polynomial by a Monomial,[object Object],To divide a polynomial by a monomial, each term is divided by that monomial. Example: (40x2- 15x + 5) ÷ 5 ,[object Object],[object Object]

We know that when we add or subtract fractions with a common denominator, we merely add or subtract the numerators, and the denominator remains the same.So we can rewrite the problem with each term over that common denominator.,[object Object],[object Object],To check: Use distributive property to multiply: ,[object Object],5(8x2 – 3x +1) = 40x2 – 15x + 5,[object Object]

Divide a Polynomial by a Monomial,[object Object],Here is another example dividing a polynomial by a monomial where the monomial divisor contains a variable: ,[object Object],(12x3 + 24x2 - 8x) ÷ 4x,[object Object],[object Object]

Again, we can rewrite the problem with each term over the common denominator.

Then we simplify each term to get the final resultTo check: Use distributive property to multiply: ,[object Object],4x(3x2 + 6x - 2) = 12x3 + 24x2 – 8x,[object Object]

If we are dividing a polynomial by another polynomial, it does not help us to rewrite each term over the divisor. For example, to rewrite (2x2 + x - 3) ÷ (x - 1) as only makes the problem more complicated.,[object Object],As an alternative, we can use long division in the same way it is used for division of real numbers. We write the problem using the division symbol .,[object Object],The first step in dividing is to look at the first term of the divisor (x) and determine how many times it divides into the first term of the dividend, i.e. what would we multiply by x to get 2x2? The result (2x) is written over the x term in the dividend ,[object Object],Divide a Polynomial by a Polynomial,[object Object],quotient,[object Object],divisor,[object Object],2x,[object Object],x – 1 2x2 + x – 3,[object Object],dividend,[object Object]

Divide a Polynomial by a Polynomial,[object Object],2x ,[object Object],x – 1 2x2 + x – 3,[object Object],2x2 – 2x,[object Object],Next we multiply the partial quotient (2x) by the divisor ,[object Object],(x - 1), and place the result below the dividend, lining up the terms ,[object Object],Then we subtract that result from the corresponding terms in the dividend and bring down the next term. Note that we are subtracting negative 2x which is the same as adding a positive 2x,[object Object],2x ,[object Object],x – 1 2x2 + x – 3,[object Object], -(2x2 – 2x),[object Object],3x – 3,[object Object]

Divide a Polynomial by a Polynomial,[object Object],2x + 3,[object Object],x – 1 2x2 + x – 3,[object Object], -(2x2 – 2x),[object Object], 3x - 3,[object Object],Next, we determine how many times x divides into 3x and place the result (3) over the constant term in the dividend.,[object Object],Multiply the 3 by (x – 1) and place the result under the dividend and subtract. We get a remainder of 0 and there are no more terms, so our result is 2x +3.,[object Object],2x + 3,[object Object],x – 1 2x2 + x – 3,[object Object], -(2x2 – 2x),[object Object], 3x - 3,[object Object], -(3x - 3),[object Object], 0,[object Object]

Check the Answer,[object Object],To check, use the FOIL method to multiply.,[object Object], (x – 1) • ( 2x + 3),[object Object],First: (x) • (2x) = 2x2,[object Object],Outer: (x) • (3) = 3x,[object Object],Inner: (-1) • (2x) = -2x,[object Object],Last: (-1) • (3) = -3,[object Object],Result: 2x2 + 3x – 2x – 3 = 2x2 + x - 3,[object Object]

Division of Polynomials,[object Object],Another example: (16z3 + 7 – 4z2) ÷ (2z -1) ,[object Object],Before we can start the division process we need to rearrange the terms in the dividend, so that they are in descending order of powers. Also note that we have z3 and z2 terms, but no z term. We will place a z term in our dividend with a coefficient of 0 to use as a placeholder. ,[object Object],We have rearranged the terms in the dividend, and found our first partial quotient of 8z2, since 2z • 8z2 = 16z3 .,[object Object],Next we multiply 8z2 by (2z -1) and place the result below the dividend lining up the terms with like powers.,[object Object], 8z2,[object Object],2z – 1 16z3 – 4z2 + 0z +7,[object Object],16z3 – 8z2 ,[object Object]

Division of Polynomials,[object Object],Next subtract the first partial product; we get a result of 4z2 and bring down the 0z. ,[object Object],Performing the next division: Since 2z • 2z gives us 4z2, we write the 2z in the z term position in the quotient.,[object Object],Now we multiply the 2z by our divisor, write the result below, and perform another subtraction.,[object Object], 8z2 + 2z,[object Object],2z – 1 16z3 – 4z2 + 0z +7,[object Object], -(16z3 – 8z2 ),[object Object],4z2 + 0z,[object Object], 8z2 + 2z,[object Object],2z – 1 16z3 – 4z2 + 0z + 7,[object Object], -(16z3 – 8z2 ),[object Object], 4z2 + 0z,[object Object],-(4z2 - 2z),[object Object], 2z,[object Object]

Division Of Polynomials

Division Of Polynomials

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Division Of Polynomials