Polynomial Division
The area of a recording studio is $3x^2 + 15x$ and its width is $3x$. How long is it? We need to divide polynomials. The division works just like the one you learned in elementary school, and it has a very fast shortcut: Ruffini's rule, also called synthetic division.
Dividing polynomials
By a monomial: divide each term. $\dfrac{3x^2 + 15x}{3x} = \dfrac{3x^2}{3x} + \dfrac{15x}{3x} = x + 5$.
By a polynomial: it's done like dividing numbers, term by term:
Divide the following polynomial by the given monomial, simplifying each of the resulting terms: $$(12x^4 - 8x^3 + 4x^2) : (4x^2)$$
Divide the following polynomial by the given monomial, simplifying each of the resulting terms:
$$(12x^4 - 8x^3 + 4x^2) : (4x^2)$$The area of a rectangular recording studio is given by the polynomial $A(x) = 3x^2 + 15x$. If the width of the studio measures $3x$ meters, what is the algebraic…
The area of a rectangular recording studio is given by the polynomial $A(x) = 3x^2 + 15x$. If the width of the studio measures $3x$ meters, what is the algebraic expression that represents the length of the studio?
Use polynomial long division to find the quotient $C(x)$ and the remainder $R(x)$ of the following polynomial operation. Remember to order and complete the terms of the…
Use polynomial long division to find the quotient $C(x)$ and the remainder $R(x)$ of the following polynomial operation. Remember to order and complete the terms of the dividend before starting:
$(6x^4 + 7x^3 - 5x^2 + 1) : (3x^2 - x + 2)$
Ruffini's rule (synthetic division)
When the divisor has the form $x - a$, there is a shortcut that uses only the coefficients. Let's divide $x^3 - 5x^2 + 3x - 4$ by $x - 2$ (so $a = 2$):
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Write the coefficients (with 0 in the gaps) and the number $a$ on the left.
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Bring down the first coefficient.
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Multiply it by $a$, write it under the next coefficient, and add.
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Repeat until the end. The last number is the remainder; the others are the coefficients of the quotient (one degree lower).
Apply Ruffini's rule to calculate the quotient and the remainder of the following division: $$(x^3 - 5x^2 + 3x + 9) : (x - 2)$$
Apply Ruffini's rule to calculate the quotient and the remainder of the following division:
$$(x^3 - 5x^2 + 3x + 9) : (x - 2)$$Use Ruffini's rule to divide the polynomial $P(x) = x^3 + 4x^2 - x - 4$ by the binomial $d(x) = x + 1$. Clearly state the quotient $C(x)$ and the remainder $R$.
Use Ruffini's rule to divide the polynomial $P(x) = x^3 + 4x^2 - x - 4$ by the binomial $d(x) = x + 1$. Clearly state the quotient $C(x)$ and the remainder $R$.
Apply the Ruffini rule to calculate the quotient and the remainder of the division of the polynomial $P(x) = x^4 - 3x^2 + 5x - 10$ by the binomial $(x + 3)$. Remember…
Apply the Ruffini rule to calculate the quotient and the remainder of the division of the polynomial $P(x) = x^4 - 3x^2 + 5x - 10$ by the binomial $(x + 3)$.
Remember that, to apply the rule, the polynomial must be complete and ordered (if any term is missing, its coefficient is written as $0$) and that the value placed on the left is the root of the divisor.
- a) Write the coefficients of the dividend and the value of $a$ that you use in the Ruffini rule.
- b) Carry out the Ruffini rule step by step.
- c) Write the quotient polynomial $C(x)$ and clearly state the value of the remainder $R$.
The remainder theorem
In the example, the remainder of dividing by $x - 2$ was $-10$. Calculate $P(2) = 8 - 20 + 6 - 4 = -10$. It matches! That's no coincidence:
This way you can find the remainder without dividing, just by substituting. And in particular:
- If $P(a) = 0$, the division by $(x - a)$ is exact: $a$ is a root and $(x - a)$ is a factor of $P(x)$.
Without carrying out long division or using Ruffini's rule, calculate the value of the remainder of the following operation using the Remainder Theorem: $$(x^4 - 3x^2 +…
Without carrying out long division or using Ruffini's rule, calculate the value of the remainder of the following operation using the Remainder Theorem:
$$(x^4 - 3x^2 + 5x - 2) : (x - 1)$$Calculate the value of the letter $k$ so that the division is exact (that is, the remainder is $0$): $$(x^2 - 6x + k) : (x - 4)$$
Calculate the value of the letter $k$ so that the division is exact (that is, the remainder is $0$):
$$(x^2 - 6x + k) : (x - 4)$$Without carrying out the division, use the Remainder Theorem to find out whether the number $x = 2$ is a root of the polynomial $P(x) = 2x^3 - 5x^2 - x + 6$. Justify…
Without carrying out the division, use the Remainder Theorem to find out whether the number $x = 2$ is a root of the polynomial $P(x) = 2x^3 - 5x^2 - x + 6$. Justify your answer by indicating whether the division of $P(x)$ by $(x - 2)$ has no remainder (is exact) or not.
Roots and factoring
Factoring means writing a polynomial as a product of simpler polynomials, just like $12 = 2^2 \times 3$. Tools, in this order:
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Common factor: $x^3 - 4x = x(x^2 - 4)$.
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Special products: $x^2 - 4 = (x + 2)(x - 2)$ (difference of squares).
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Ruffini testing integer roots: if the polynomial has integer coefficients, its integer roots are among the divisors of the constant term.
Algebraic fractions
A fraction with polynomials is simplified like fractions of numbers: by factoring the top and bottom and canceling common factors.
$$\frac{x^2 - 9}{x^2 + 3x} = \frac{(x + 3)(x - 3)}{x\,(x + 3)} = \frac{x - 3}{x}\qquad (x \ne 0,\ x \ne -3)$$The restrictions come from the original denominator: $x(x + 3)$ cannot equal zero. Even though the factor $x + 3$ disappears, $x = -3$ is still excluded. The simplified expression matches the original one only on its original domain.
Check whether the number $a = 3$ is a root of the polynomial $P(x) = x^3 - 3x^2 + x - 3$ by carrying out division using Ruffini's rule and observing the remainder…
Check whether the number $a = 3$ is a root of the polynomial $P(x) = x^3 - 3x^2 + x - 3$ by carrying out division using Ruffini's rule and observing the remainder obtained. Justify your answer based on the result of the operation.
Check whether the binomial $(x + 1)$ is a factor of the polynomial $P(x) = x^4 - 1$. To do this, you can use long division, Ruffini's rule, or the Factor Theorem.…
Check whether the binomial $(x + 1)$ is a factor of the polynomial $P(x) = x^4 - 1$. To do this, you can use long division, Ruffini's rule, or the Factor Theorem. Explain what it means for a binomial to be a "factor" of another polynomial in terms of the remainder of the division.
The area of a rectangular padel court is given by the polynomial $A(x) = 2x^2 + 11x + 12$. If the width of the court measures $(x + 4)$ meters, determine the algebraic…
The area of a rectangular padel court is given by the polynomial $A(x) = 2x^2 + 11x + 12$. If the width of the court measures $(x + 4)$ meters, determine the algebraic expression that represents its length.