How To Find The Remaining Zeros Of A Polynomial

A new bakery offers decorated cakes for children’s birthday parties and other special occasions. The bakery wants the volume of a small cake to be 351 cubic inches. The cake is rectangular in shape. They wanted the length of the cake to be four inches longer than the width of the cake, and the height of the cake to be one-third the width. What size should the cake be? This problem can be solved by writing a cubic function and solving the cubic equation for the volume of the pie. In this section, we will discuss many tools for writing polynomial functions and solving polynomial equations.

Theorems used to analyze polynomial functions

In the previous section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using The remaining theorem. If the polynomial is divided by x – k, the remainder can be found quickly by evaluating the polynomial function at k, that is, f(k) Let’s go through the proof of the theorem. Division Algorithm The statement that, given a polynomial divisor f(x) and a non-zero divisor of the polynomial d(x) where the degree of d(x) is less than or equal to the degree of f(x), exists unique polynomial q(x) and r(x) such that[latex]fleft (xright) = dleft (xright) qleft (xright) + rleft (xright)[/latex]If the divisor, d(x), is x – k, then the number is of the form[latex]fleft (xright) = left (x-kright) qleft (xright) + r[/latex]Since the divisor x – k is linear, the remainder will be a constant, r. And, if we evaluate this for x = k, we have[latex]begin {array} {l} fleft (kright) = left (k-kright) qleft (kright) + rhfill text {} = 0cdot qleft (kright) + rhfill text {} = rhfill end {array}[/latex]In other words, f(k) is the remainder obtained when f(x) is divided by x – k.

Use Rational Zero Theorem to find rational zeros

Read more: How to reuse the mx cherry stabilizer without removing the key But first we need a group of rational numbers to check. The Rational Zero’s Theorem help us narrow down the number of possible irrational numbers by using the ratio between the factors of the constant term and the factors of the leading number coefficient of polynomials Consider a quadratic function with two zeros, [latex]x = fraction {2} {5}[/latex] and [latex]x = fraction {3} {4}[/latex]According to the Coefficients Theorem, these zeros have coefficients associated with them. Let’s set each factor to zero, and then construct the original quadratic function without its stretching factor. Note that two of the factors of the constant term, 6, are two numerators from the rational roots: 2 and 3. Similarly, two of the factors from the original rational coefficient, are two denominators from rational roots: 5 and 4 We can infer that the numerator of the rational roots will always be a factor of the constant term and the denominator will be the factor of the leading coefficient . This is the essence of Rational Zero’s Theorem; it is a means to give us a group of possible logical zeros.

Use the Factoring Theorem to solve polynomial equations

The Factorial theorem is another theorem that helps us to analyze polynomial equations. It tells us how the zeros of a polynomial are related to the factors. Recall that the Algorithm divides us[latex]fleft (xright) = left (x-kright) qleft (xright) + r[/latex].If k is 0, then the remainder r is [latex]Fleft (kright) = 0[/latex] and [latex]fleft (xright) = left (x-kright) qleft (xright) +0[/latex] or [latex]fleft (xright) = left (x-kright) qleft (xright)[/latex]Note, written in this form, x – k is the coefficient of [latex]Fleft (xright)[/latex]. We can conclude if k is zero of [latex]Fleft (xright)[/latex]afterward [latex]xk[/latex] is an element of [latex]Fleft (xright)[/latex]Similarly, if [latex]xk[/latex] is an element of [latex]Fleft (xright)[/latex]then the rest of the Division Algorithm [latex]fleft (xright) = left (x-kright) qleft (xright) + r[/latex] is 0. This tells us that k is 0. Read more: Lesbian Fashion – Hottest Lesbian Outfits for 2021 This pair of implication is the Factor Theorem. As we will see shortly, a polynomial of degree n in complex coefficients will have n zeros. We can use the Factorial Theorem to completely multiply a product polynomial of n factors. Once the polynomial is completely producted, we can easily determine the zeros of the polynomial.

Finding Zeros of a Polynomial Function

The Rational Zero theorem helps us narrow down the list of possible rational zeros for a polynomial function. Once we’ve done this, we can use synthetic division repeat to determine all zero of a polynomial function.

Basic Theorem of Algebra

Now that we can find rational zeros for a polynomial function, we will consider a theorem that discusses the number of complex zeros of a polynomial function. The Basic Theorem of Algebra tells us that every polynomial function has at least one complex zero. This theorem forms the basis for solving polynomial equations. Suppose f is a polynomial of order 4, and [latex]fleft(xright) = 0[/latex]. The Fundamental Theorem of Algebra states that there is at least one complex solution, let’s call it [latex]{c} _ {1}[/latex]. According to the Factorial Theorem, we can write [latex]Fleft (xright)[/latex] as a product of [latex]x- {c} _ {text {1}}[/latex] and a polynomial quotient. Are from [latex]x- {c} _ {text {1}}[/latex] is linear, the quotient of the polynomial will be cubic. We now apply the Fundamental Theorem of Algebra to the third degree polynomial quotient. It will have at least one complex zero, let’s call it [latex]{c} _ {text {2}}[/latex]. So we can write the quotient of the polynomial as a product of [latex]x- {c} _ {text {2}}[/latex] and a new polynomial quotient of the second degree. Keep applying the Fundamental Theorem of Algebra until all zeros are found. There will be four of them and each will yield a multiplier [latex]Fleft (xright)[/latex].

Linear Factorization and Descartes’ Rule of the Sign

An important implication of Basic Theorem of Algebra is a polynomial of order n that will have n zeros in the set of complex numbers, if we allow multiplication. This means that we can multiply the polynomial function by n factors. The Linear Factorial Theorem tells us that a polynomial function will have the same number of factors as its degree and that each factor will have the form (x – c), where c is a complex number. suppose [latex]a + bitxt {,} bne 0[/latex]is the zero of [latex]Fleft (xright)[/latex]. Then, according to the Coefficients Theorem, [latex]x-left (a + biright)[/latex] is an element of [latex]Fleft (xright)[/latex]. For f to have a real coefficient, [latex]x-left (a-biright)[/latex] must also be an element of [latex]Fleft (xright)[/latex]. This is true because any factor other than [latex]x-left (a-biright)[/latex]when multiplying by [latex]x-left (a + biright)[/latex], will leave imaginary ingredients in the product. Only multiplication by conjugate pairs removes the imaginary parts and results in real coefficients. In other words, if a polynomial function f with real coefficients has a complex number of 0 [latex]a + bi[/latex]then the complex conjugate [latex]a-bi[/latex] must also be 0 [latex]Fleft (xright)[/latex]. This is called Complex Conjugation Theorem.

Descartes’s Rule of the Sign

There is a simple way to determine the number of possible positive and negative non-real numbers for any polynomial function. If the polynomial is written in descending order, Descartes’s Rule of the Sign tells us about the relationship between the number of sign changes in [latex]Fleft (xright)[/latex] and the number of positive non-real numbers. For example, the polynomial function below has a variable sign, which tells us that the function must have a positive real zero. [latex]Fleft (-right)[/latex] and the number of negative non-real numbers. In this case, [latex]fleft (correct mathrm {-x})[/latex] There are 3 signs of change. This tells us that [latex]Fleft (xright)[/latex] There can be 3 or 1 negative non-real numbers.

Solve real world applications of polynomial equations

We have now introduced a series of tools for solving polynomial equations. Use these tools to solve the bakery problem at the beginning of this section. Read more: How to paint wicker furniture with chalk paint

Gist

• To find [latex]skim (kright)[/latex] determine the remainder of the polynomial [latex]Fleft (xright)[/latex] when it is divided by [latex]xk[/latex].
• k is the zero of [latex]Fleft (xright)[/latex] if and only if [latex]left (x-kright)[/latex] is an element of [latex]Fleft (xright)[/latex] .
• Every rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient.
• When the leading coefficient is 1, the possible rational zeros are factors of the constant term.
• Composite division can be used to find the zeros of a polynomial function.
• According to the Fundamental Theorem, every polynomial function has at least one complex zero.
• Every polynomial function with degree greater than 0 has at least one complex zero.
• For multiplication, a polynomial function will have the same number of factors as its degree. Each element will have the form [latex]left (x-cright)[/latex] where c is a complex number.
• The number of positive non-reals of a polynomial function is the number of sign changes of the function or less than the number of sign changes of an even integer.
• The number of negative real zeros of a polynomial function is the number of times the sign of . is changed [latex]Fleft (-right)[/latex] or less the number of signs changed by an even integer.
• Polynomial equations model many real-world situations. Solving equations is most easily done by compound division.

glossary

Descartes’s Rule of the Sign a rule that determines the largest possible number of positive and negative non-reals based on the number of times the sign of [latex]Fleft (xright)[/latex] and [latex]Fleft (-right)[/latex] Factorial theorem k is the zero of the polynomial function [latex]Fleft (xright)[/latex] if and only if [latex]left (x-kright)[/latex] is an element of [latex]Fleft (xright)[/latex] Basic Theorem of Algebra a polynomial function of degree greater than 0 has at least one complex zero Linear Factorial Theorem for multiplication, a polynomial function will have the same number of factors as its degree, and each factor will have the form [latex]left (x-cright)[/latex] where c is a complex number Rational Zero’s Theorem possible rational zeros of a polynomial function of the form [latex]fraction {p} {q}[/latex] where p is the coefficient of the constant term and q is the factor of the leading coefficient. The remaining theorem if a polynomial [latex]Fleft (xright)[/latex] divided by [latex]xk[/latex] then the remainder is equal to [latex]skim (kright)[/latex]

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