Roots of polynomials are the solutions for any given polynomial for which we need to find the value of the unknown variable. If we know the roots, we can evaluate the value of the polynomial to zero.
An expression of the form anxn + an-1xn-1 + …… + a1x + a0, where each variable has a constant accompanying it as its coefficient is called a polynomial of degree ‘n’ in variable x. Each variable separated with an addition or subtraction symbol in the expression is better known as the term.
The degree of the polynomial is defined as the maximum power of the variable of a polynomial.
For example, a linear polynomial of the form ax + b is called a polynomial of degree 1. Similarly, quadratic polynomials and cubic polynomials have a degree of 2 and 3 respectively.
A polynomial with only one term is known as a monomial. A monomial containing only a constant term is said to be a polynomial of zero degrees. A polynomial can account for a null value even if the values of the constants are greater than zero. In such cases, we look for the value of variables that set the value of the entire polynomial to zero. These values of a variable are known as the roots of polynomials. Sometimes they are also termed as zeros of polynomials.
Roots of Polynomials Formula
The polynomials are the expression written in the form of:
anxn+an-1xn-1+……+a1x+a0
The formula for the root of a linear polynomial such as ax + b is
x = -b/a
The general form of a quadratic polynomial is ax2 + bx + c and if we equate this expression to zero, we get a quadratic equation, i.e. ax2 + bx + c = 0.
The roots of a quadratic equation, whose degree is two, such as ax2 + bx + c = 0 are evaluated using the formula;
x = [-b ± √(b2 – 4ac)]/2a
The formulas for higher-degree polynomials are a bit complicated.
Roots of three-degree polynomial
To find the roots of the three-degree polynomial we need to factorize the given polynomial equation first so that we get a linear and quadratic equation. Then, we can easily determine the zeros of the three-degree polynomial. Let us understand with the help of an example.
Example: 2x3 − x2 − 7x + 2
Divide the given polynomial by x – 2 since it is one of the factors.
2x3 − x2 − 7x + 2 = (x – 2) (2x2 + 3x – 1)
Now we can get the roots of the above polynomial since we have got one linear equation and one quadratic equation for which we know the formula.
Finding Roots of Polynomials
Let us take an example of the polynomial p(x) of degree 1 as given below:
p(x) = 5x + 1
According to the definition of roots of polynomials, ‘a’ is the root of a polynomial p(x), if
P(a) = 0.
Thus, in order to determine the roots of the polynomial p(x), we have to find the value of x for which p(x) = 0. Now,
5x + 1 = 0
x = -1/5
Hence, ‘-1/5’ is the root of the polynomial p(x).
Questions and Solutions
Example 1: Check whether -2 is a root of polynomial 3x3 + 5x2 + 6x + 4.
Solution: Let the given polynomial be,
p(x) = 3x3 + 5x2+ 6x + 4
Substituting x = -2,
p(-2) = 3(-2)3+ 5 (-2)2 + 6(-2) + 4
p(-2) = -24 + 20 – 12 + 4 = -12
Here, p(-2) ≠ 0
Therefore, -2 is not a root of the polynomial 3x3 + 5x2 + 6x + 4.
Example 2: Find the roots of the polynomial x2 + 2x – 15
Solution: Given x2 + 2x – 15
By splitting the middle term,
x2 + 5x – 3x – 15
= x(x + 5) – 3(x + 5)
= (x – 3) (x + 5)
⇒ x = 3 or x =−5
Basic theoretical information about counting Degree and roots.
Some recommendations for algebraic computations, transformations, and simplifications.
- Convert decimal fractions to ordinary ones, i.e. those who have a numerator and denominator.
- Do not try to count the whole expression at once. Perform calculations on a single action, step by step. In this case, note that:
- first perform operations in brackets;
- then count the works and / or divisions;
- then add up or subtract;
- and lastly, if it was a multi-story fraction, divide the already completely simplified numerator into a completely simplified denominator too;
- moreover, performing first operations in brackets also observe the same sequence, first works or divisions inside brackets, then summation or subtraction in brackets, and if there is another bracket inside the bracket, then the actions in it are performed first of all.
- Do not rush to multiply and divide the “terrible numbers.” Most likely, in one of the following actions something will be reduced. To make it easier to reduce the numbers can be decomposed into prime factors.
- When adding and subtracting, select the whole part in fractions (if possible). When multiplying and dividing, on the contrary, reduce the fraction to a form without an integer part.
From the roots in the denominator taken to get rid of. To get rid of the root over the whole denominator, multiply the numerator and denominator by an expression equal to the denominator. To get rid of the root over a part of the denominator, multiply the numerator and denominator by the conjugate denominator expression. In this case, the difference of squares is formed (conjugate for ( a – b ) is the expression ( a + b ) and vice versa).
When transforming or simplifying algebraic expressions, the sequence of actions is as follows:
- Factorize everything that can be factored out.
- Cut all that can be cut.
- And only then lead to a common denominator. In no case do not try at once headlong to lead to a common denominator. An example will become the farther, the worse.
- Again factor out and cut.
In order to convert a decimal periodic fraction to an ordinary fraction (with a numerator and denominator), you must:
- From the number that stands up to the second period in the original periodic fraction, subtract the number that stands up to the first period in the same fraction and record the difference obtained in the numerator of the future ordinary fraction.
- In the denominator, write as many nines as there are digits in the period of the initial fraction, and as many zeros as there are digits between the comma and the first period.
- Do not forget about the whole part, if it is.
When solving problems from this topic, it is also necessary to remember a lot of information from previous topics. Here are the main ones.
Abbreviated Multiplication Formulas
When performing various algebraic transformations, it is often convenient to use abbreviated multiplication formulas. Often these formulas are used not so much to shorten the multiplication process, but on the contrary, rather to see from the result that it can be represented as a product of some multipliers. Thus, these formulas need to be able to apply not only from left to right, but also from right to left. We list the basic formulas of abbreviated multiplication:
The latter two formulas are also often conveniently used in the form:
Square trinomial and Vieta theorem
In the case when a quadratic equation has two roots, the corresponding quadratic term can be factorized by the following formula :
If the quadratic equation has one root, then the decomposition of the corresponding square trinomial into factors is given by the following formula :
Only if the quadratic equation has two roots (that is, the discriminant is strictly greater than zero) the Viet theorem holds . According to the Vieta Theorem, the sum of the roots of a quadratic equation is:
The product of the roots of a quadratic equation according to the Viet theorem can be calculated by the formula:
So again about the theorem of Viet:
- If D <0 (the discriminant is negative), then the equation has no roots and the Vieta theorem cannot be applied.
- If D > 0 (the discriminant is positive), then the equation has two roots and the Viet theorem works fine.
- If D = 0, then the equation has a single root, for which it is meaningless to introduce the concept of a sum or a product of roots, therefore the Viet theorem is also not applicable.
Basic properties of degrees
Mathematical degrees have several important properties; we list them:
The last property holds only for n > 0. Zero can only be raised to a positive power. Well, the main property of the negative degree is written as follows:
Basic properties of mathematical roots
The mathematical root can be represented in the form of an ordinary degree, and then use all the properties of the degrees given above. To represent the mathematical root in the form of a degree, use the following formula:
Nevertheless, it is possible to write out separately a number of properties of mathematical roots, which are based on the properties of the degrees described above:
For arithmetic roots, the following property holds (which can also be considered the definition of a root):
The latter is true: if n is odd, then for any a ; if n is even, then only if a is greater than or equal to zero. For the root of odd degree , the following equality is also fulfilled (from the root of the odd degree, one can remove the minus sign):
Since the value of a root of even degree can only be non-negative , for such roots there is the following important property:
So you should always remember that under the root of an even degree only a non-negative expression can stand, and the root itself is also a non-negative expression. In addition, it should be noted that if you use a record with a mathematical root icon, then the exponent of this root can only be an integer, and this number must be greater than or equal to two:
Basic properties of square root
The square root is the second degree mathematical root:
A square root can only be extracted from a non-negative number. The value of the square root is also always non-negative:
For a square root, there are two important properties that are important to remember very well and not to confuse:
If there are several factors under the root, then the root can be extracted from each of them separately. It is important to understand that each of these factors separately (and not just their product) must be non-negative:
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