Exams Prep Master
Polynomials are expressions commonly found in Algebra. These algebraic expressions consist of two major parts, variables and coefficients. These take part only in operations like addition, subtraction, multiplication, and exponentiation of non-negative integers. For example, a polynomial having a single variable or an indeterminate x is given as 2x2 - 8x + 14. The word polynomial is derived from two major roots. The Greek term “poly”, which means many, and the Latin term “nomen”, signifies terms . So polynomial means many or multiple number of terms.
The video below explains this:
Polynomials Detailed Video Explanation:
Very Short Answer Questions [1 Mark Questions]
Ques. What is a monomial?
Ans. A monomial is basically a type of polynomial which has only one term. Therefore the phrase “mono” is used. Although it comprises only a single term, it can easily take higher degrees along with multiple variables in an algebraic expression. They are highly essential in Mathematics for carrying out different analytical problems.
Ques. Give an example of a monomial and a binomial having degrees as 40 and 55 respectively.
Ans. For example, 6abc6, happens to be a single term, where 6 is the coefficient of the expression, a, b, c are the different variables present in it, whereas 6 is the degrees of the monomial. A monomial having a degree of 40 is x40, whereas, a binomial having a degree of 55 is written as x55 + x.
Ques. A polynomial called a is present. How many terms will it have if it has a degree 4?
Ans. If a polynomial exists with a degree of 4, then it will consist of 4 terms. For example, a4 means 4 times a, or it can be broken down as a × a × a × a. Similarly, if a polynomial x with a degree 2 exists, it basically denotes that it will have a total of 2 terms.
Ques. What is the coefficient of x2 in the given polynomial x2 + 2y4 + z1?
Ans. It is clearly evident that the coefficient of x2 in the given polynomial x2 + 2y4 + z1 is 1. It is hidden but it basically means that 1 × x2. It is applicable to every other variable, for example, the coefficient of y4 will be 2, as it is in the form of 2y4.
Ques. What are the factors of the equation x2 – x – 6?
Ans. The equation is given as x2 – x – 6
x2 – 3x + 2x – 6
x(x - 3) + 2(x - 3) [taking x and 2 common from both respectively]
(x + 2) (x - 3)
Thus, the two factors are (x + 2) (x - 3).
Ques. What are the different types of polynomials?
Ans. There are polynomials of different nature like monomials, binomials, and trinomials. Depending upon their unique characteristic, they can be summarised into 4 types called as zero, linear, cubic, and quadratic polynomials respectively.
Short Answer Questions [2 Marks Questions]
Ques. The equation 2x2 − 4ax + 2a − 7 has a potential factor (x + 1). What is the possible value of the expression?
Ans. As x + 1 is a factor, we can easily substitute the value of x as -1 in the equation because x + 1 = 0, so x = -1
Now,
P(x) = 2 (-1)2 – 4a(-1) + 2a
p(-1) = 2 + 4a + 2a
p(-1) = 2 + 6a
Or
0 = 2 + 6a
6a = -2
a = -2/6
a = -1/3
Ques. In a quadratic polynomial of the given form x2 + ax + b, if one of the zeroes (α) happens to be negative of the other, then what is its nature?
Ans. It has no linear term and the constant term becomes negative.
We know that sum of the zeroes is,
⇒ α + (-α) = -a
⇒ a = 0
Product of the zeroes will be,
⇒ α(-α) = b
⇒ b = -a2
b will ultimately become negative on the number line. Hence, the polynomial will not possess any linear term, and the constant term will become negative.
Ques. Find the remainder when a3 + a2 + a + 1 is divided by a – 3/2.
Ans. According to the Remainder Theorem, we know that,
p(a) = a3 + a2 + a + 1, and q(a) = a – 3/2
We know that q(a) will divide p(a)
By using the theorem, we get,
p(3/2) = (3/2)3 + (3/2)2 + (3/2) + 1
Thus, 27/8 + 9/4 + 3/2 + 1
= (27 + 18 + 12 + 8) / 8
= 65/8
The remainder of the given equation is \(\frac{65}{8}\).
Ques. The base side of a triangle is given as 2x2 + 2x – 12. Base is the same as the height. What will be the area of the triangle?
Ans. Given that base of the triangle = 2x2 + 2x - 12
Thus, by simplifying the equation through factoring, we can find out the area.
2x2 + 2x - 12
= 2 (x2 + x - 6) [taking 2 common from all]
= 2( x2 + 3x – 2x – 6)
= 2{x(x + 3) – 2(x + 3)}
= 2 (x – 2) (x + 3)
Thus base and height will be 2 (x – 2) (x + 3)
Area of triangle = ½ × 2(x-2) (x+3) × 2(x-2) (x+3)
Area = 2 (x-2)2 (x+3)2
Ques. Given that one of the zeroes of the mentioned cubic polynomial x3 + ax2 + bx + c is –1, then what will be the product of the other two zeroes present?
Ans. according to formula, we know that
P(x) = x3 + ax2 + bx + c
Now, one of the zero is given as -1, therefore upon substituting the value of x as -1, we get,
P(-1) = (-1)3 + a(-1)2 + b(-1) + c
0 = -1 + a – b + c
b – a + 1 = c
By formula, product of all zeroes is
αβγ = \(\frac{-\text { constant term }}{\text { coefficient of } x^{3}}\)
⇒ (-1) βγ = \(\frac{-c}{1}\)
βγ = c
βγ = 1 – a + b
Long Answer Questions [3 Marks Questions]
Ques. There are two different equations given below.
2x2 + 16x + 30 —(i)
2x2 + 6x – 20 —(ii)
Calculate the common factor in them.
Ans. Upon simplifying the first equation,
2x2 + 16x + 30 = 0
2 (x2 + 8x + 15) = 0 [taking 2 common]
2 [x2 + 5x + 3x + 15]
2(x+3)(x+5)
Upon taking the second equation,
2x2 + 6x – 20 = 0
2 (x2 + 3x – 10) = 0 [taking 2 common]
2 [ (x2 + 5x – 2x – 10) ]
2 [ x(x + 5) – 2(x + 5) ]
2(x+5)(x-2)
Thus, the common factor present in both of them is (x+5).
Ques. What will be the value of the polynomial 10x – 8x2 + 6 at x = 4 and at x = -1?
Ans. Let us assume the polynomial to be f(x) = 10x – 8x2 + 6
Case 1 – For x = 4 (Substituting 4 in the place of x)
f(4) = 10 × 4 – 8 × 42 + 6
f(4) = 40 – 8 × 16 + 6
f(4) = -82
The value of the polynomial 10x - 8x2 + 6 at x = 4 is -82
Similarly, case 2, for x = -1 (Substituting -1 in the place of x)
f(-1) = 10 × (-1) – 8 × (-1)2 + 6
f(-1) = -10 – 8(1) + 6
f(-1) = -10 – 2
f(-1) = -12
Thus the two values are -82 and -12.
Ques. If ab = 8, and 3a + 2b = 15, then what will be the value of 9a2 + 4b2?
Ans. Let us take the equation 3a + 2b = 15
Upon squaring on both sides, we would get,
(3a + 2b)2 = (15)2
9a2+ 4b2+12ab = 225
9a2+ 4b2 = 225 – 12ab
We already know that the value of ab is 8. Herefore, upon substituting it we get
9a2 + 4b2 = 225 – 12 × 8
9a2 + 4b2 = 225 – 96
9a2 + 4b2 = 129
Thus the value of 9a2 + 4b2 = 129
Very Long Answers Questions [5 Marks Questions]
Ques. The area of a given rectangular table is 7x2 – 6 = 19x. What will be its perimeter?
Ans. Given that,
Area of rectangle = 7x2 – 6 = 19x.
Area of a rectangle = length × breadth
We know that by factoring, we can break a quadratic equation and thus split it. This in turn will give us length and breadth for the rectangle.
Thus, upon factoring, we get the following result,
7x2 – 6 – 19x = 0
7x2 – 19x – 6 = 0
7x2 – 21x + 2x – 6 = 0
7x (x – 3) + 2 (x – 3) = 0 (Taking 7x common)
(x-3)(7x+2)
Therefore, the length and breadth for the given rectangle are (x-3) (7x+2).
Now, according to formula, we know that
Perimeter of a rectangle = 2(length + breadth)
Therefore,
Perimeter of the rectangular table = 2[(x - 3) + ( 7x + 2)]
= 2 [ 8x – 1 ]
= 16x – 2
= 2(8x – 1)[taking 2 common]
Ques. What is the Remainder theorem of a polynomial? What will be the result if 4x3 – 5x + 1 is divided by x – 2 and x – 1?
Ans. The Remainder theorem states that if a polynomial p(x) is divided by a binomial/linear polynomial x-a, then the resultant remainder will be p(a). It should be noted that x-a is the divisor of p(x), only if p(x)=0. The theorem is also known as little Bézout's theorem. This theorem helps us to connect the remainder, along with the dividend. The main motive of this is to find out the remainder, without doing the actual division.
Case 1
When the divisor is x - 2
x - 2 = 0
Therefore x = 2
Substituting the value of x in the given equation
Let f(x) = 4x3 – 5x + 1
R = f(2) = 4(2)3 – 5(2) + 1
= 4 × 8 – 10 + 1
= 32 – 9
= 23
Case 2
When the divisor is x - 1
x - 1 = 0
Therefore x = 1
Substituting the value of x in the given equation
Let f(x) = 4x3 – 5x + 1
R = f(1) = 4(1)3 – 5(1) + 1
4 × 1 – 5 + 1
4 – 5 + 1
-1 + 1
As both get cancelled, we are left with 0.
Thus when 4x3 – 5x + 1 is divided by x-2 and x-1, the remainders are 23 and 0 respectively.
Ques. Find the value of the equation a3 + b3 + c3 – 3abc, if a2 + b2 + c2 = 50 and a + b + c = 10.
Ans. Let us take the equation a + b + c = 10
According to rules of the algebraic equalities, we know that (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
Substituting the values in the expression
(10)2 = 50 + 2(ab +bc + ca)
100 – 50 = 2(ab +bc + ca)
50/2 = (ab +bc + ca)
(ab + bc + ca) = 25
Now, again, we know that a³ + b³ + c³ – 3abc = (a + b + c)(a² + b² + c² – ab – bc – ca)
a³ + b³ + c³ – 3abc = (a + b + c)(a² + b² + c² – ab – bc – ca)
a³ + b³ + c³ – 3abc = 10 ( 50 – 25) [substituting values]
a³ + b³ + c³ – 3abc = 10 × 25
a³ + b³ + c³ – 3abc = 250
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