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Right ANSWER is (c) x^2 – BX + c

The best EXPLANATION: α and β are the zeros of x^2 + bx + c

Also, α + β = -B and αβ = c

Now, -α-β = -(-b) = b and -α × -β = c

Hence, the POLYNOMIAL with -α, -β as its zeros will be x^2 – bx + c

Correct answer is (c) (0, 5)

The explanation: The graph of the polynomial 2x^2-8x+5 cuts the y-axis.

Hence, the value of x will be 0.

y(0)=2(0)^2-8(0)+5

y=5

The graph cuts the y-axis at (0,5)

The correct ANSWER is (b) α+β>αβ

For explanation: The given polynomial is x^2+35x-75.

The sum of ZEROS, α + β = \(\FRAC {-coefficient \, of \, x}{coefficient \, of \, x^2} = \frac {-35}{1}\) = -35

The product of zeros, αβ = \(\frac {constant \, TERM}{coefficient \, of \, x^2}\) = -75

Clearly, sum of zeros is greater than product of zeros.

The CORRECT answer is (d) a=-8, b=-5

Explanation: We know that,

f(x)=q(x)×G(x)+r(x)

Where, f(x) is the DIVIDEND, q(x) is the QUOTIENT, g(x) is the divisor and r(x) is the remainder.

∴ 30x^4-50x^3+109x^2-23x+25=(10x^2+3)(3x^2-5x+10)+ax+b

30x^4-50x^3+109x^2-23x+25=30x^4-50x^3+109x^2-15x+30+ax+b

30x^4-50x^3+109x^2-23x+25-(30x^4-50x^3+109x^2-15x+30)=ax+b

-23x+25+15x-30=ax+b

-8x-5=ax+b

∴ a=-8, b=-5

The correct option is (c) 30

For explanation I would say: 81x^2-31 is exactly DIVISIBLE by 9x+1

Hence, on dividing 81x^2-31 by 9x+1

We get, 9x-1 as QUOTIENT and REMAINDER as -30.

So if we add 30 to 81x^2-31, it will be exactly divisible by 9x+1.

The CORRECT answer is (d) -4

Explanation: α, β and γ are the ZEROS of 5x^3 + 10x^2 – x + 20

The PRODUCT of zeros or αβγ = \(\frac {-constant \, TERM}{coefficient \, of \, x^3} = \frac {-20}{5}\) = -4

Right OPTION is (c) \(\frac {B^2-2ca}{a^2}\)

For explanation: β and γ are the zeros of the POLYNOMIAL ax^3 + bx^2 + cx + d

So, α + β + γ = \(\frac {-b}{a}\)

αβ + βγ + γα = \(\frac {c}{a}\)

Now, α^2 + β^2 + γ^2 = (α + β + γ)^2 – 2(αβ + βγ + γα)

α^2 + β^2 + γ^2 = \((\frac {-b}{a})\)^2 – 2\((\frac {c}{a}) = \frac {b^2}{a^2}\) – 2 \(\frac {c}{a} = \frac {b^2-2ca}{a^2}\)

Right OPTION is (b) √x+2x+4

To elaborate: An EXPRESSION in the form of (x)=a0+a1x+a2x^2+…+anx^n, where an≠0, is CALLED a polynomial where a1, a2 … an are REAL numbers and each POWER of x is a non-negative integer.

In case of √x+2x+4, the power of x is not an integer.

Therefore it is not a polynomial.

Correct OPTION is (b) True

For explanation I would say: The DEGREE of the polynomial is the highest of the degree of the polynomial. Hence, a polynomial with highest degree ONE is linear, two as quadratic and so on.

Right answer is (C) 2x^2-20x+5

Easy explanation: The sum of the polynomial is 10, that is, α+β = 10

The product of the polynomial is \(\FRAC {5}{2}\) i.e. αβ = \(\frac {5}{2}\)

∴ f(X)=x^2-(α+β)x+αβ

f(x)=x^2-10x+\(\frac {5}{2}\)

f(x)=2x^2-20x+5

Right option is (B) x^4-90x^2+729

To explain I would say: The ZEROS of the polynomial are 3, -3, 9 and -9.

Then, (x-3), (x+3), (x-9) and (x+9) are the factors of the polynomial.

Multiplying the factors, we have

(x-3) (x+3) (x-9) (x+9)

(x^2-9) (x^2-81) (By IDENTITY (x-a)(x+a)=x^2-a^2)

(x^4-9x^2-81x^2+729)

x^4-90x^2+729

The correct option is (a) False

Explanation: The graph of the POLYNOMIAL will have a downward OPENING SINCE, a<0

The graph for the same can be OBSERVED here,

Correct choice is (C) 5-√3

To explain I WOULD SAY: SINCE the zeros of the polynomial are 3 and -2.

The divisor of the polynomial will be (x-3) and (x+2).

Multiplying (x-3) and (x+2) = x^2+2x-3x-6=x^2-x+6

Dividing, x^3+(6-√3)x^2+(-1-√3)x+30-6√3 by x^2-x+6

We get, x-5+√3 as quotient.

Hence, the third zero will be 5-√3.

Correct choice is (b) -11

The explanation is: On dividing, 15x^5+70x^4+35x^3-135x^2-40x-11 by 5x^4+10x^3-15x^2-5x

We get, 3x+8 as QUOTIENT and remainder as -11.

So if we subtract -11 from 15x^5+70x^4+35x^3-135x^2-40x-11 it will be EXACTLY DIVISIBLE by 5x^4+10x^3-15x^2-5x.

Correct CHOICE is (a) True

For explanation: The LEADING coefficient of the polynomial is LESS than zero, HENCE, it has downward opening.For example, the GRAPH of -x^2 is

Right answer is (d) -1

The best I can explain: The given polynomial is x^3 – 9x^2 – x + 9.

The TWO zeros of the polynomial are 1 and 9.

We KNOW that, the sum of zeros of the polynomial or α + β + γ = \(\frac {-COEFFICIENT \, of \, x^2}{coefficient \, of \, x^3} = \frac {9}{1}\)

1 + 9 + γ =9

γ = 9 – 10 = -1

Correct option is (a) True

To explain I would say: If the graph meets x-axis at one point only, then the QUADRATIC polynomial has COINCIDENT ZEROS. ALSO, the discriminant of the quadratic polynomial is zero, therefore roots will be real.

Right CHOICE is (C) ax^2+b

For explanation I would SAY: We know that,

f(x)=q(x)×g(x)+r(x)

Where, f(x) is the DIVIDEND, q(x) is the QUOTIENT, g(x) is the divisor and r(x) is the remainder.

acx^3 + bcx + d = cx × g(x) + d

acx^3 + bcx + d – d = cx × g(x)

\(\frac {acx^3+bcx}{cx}\)=g(x)

g(x)=ax^2+b

Right choice is (d) (3, 0) and (-5, 0)

For explanation: Since, the ZEROS of the polynomial are 3 and -5.

Therefore, x = 3 and x = -5 and they cut the x-axis so the y-coordinate will be zero.

Hence, the points it CUTS the x-axis will be (3, 0) and (-5, 0).

The CORRECT OPTION is (c) Real, Distinct

To explain: The zeros of the quadratic POLYNOMIAL cut the x-axis at TWO DIFFERENT points.

∴ b^2 – 4ac ≥ 0

Hence, the nature of the zeros will be real and distinct.

Right CHOICE is (a) 10x+2

Explanation: We know that,

f(x)=q(x)×g(x)+r(x)

Where, f(x) is the dividend, q(x) is the quotient, g(x) is the DIVISOR and r(x) is the remainder.

∴ 50x^2-90x-25=q(x)×5x-10-5

50x^2-90x-25+5=q(x)×5x-10

\(\FRAC {50x^2-90x-20}{5x-10}\)=q(x)

We GET, q(x)=10x+2

Correct choice is (B) x-α

Best explanation: If α is a zero of the polynomial F(x).

The DIVISOR will be x-α.

For EXAMPLE, if 5 is a zero of a polynomial f(x), then its divisor will be x-5.

Right choice is (a) (x^2+3)(x^2-3)

The explanation is: A BIQUADRATIC polynomial has highest POWER 4.

Hence, the polynomial with the highest power as 4 is x^4-9 or (x^2+3)(x^2-3).

The CORRECT answer is (a) \(\frac {7}{6}, \frac {1}{3}\)

The best EXPLANATION: 18x^2-27x+7=0

18x^2-21x-6x+7=0

3x(6x-7)-1(6x-7)=0

(6x-7)(3x-1)=0

x=\(\frac {7}{6}, \frac {1}{3}\)

The ZEROS are \(\frac {7}{6}\) and \(\frac {1}{3}\).

Right answer is (d) Complex

For explanation I WOULD say: Since, the graph is completely above or below the x-axis, HENCE, it has no REAL roots. If a polynomial has real roots only then it cuts the x-axis. If it lies above or below, the roots are complex in NATURE.

Right choice is (c) CUBIC

Easiest explanation: Since, the graph of the POLYNOMIAL CUTS the x-axis at 3 POINTS, hence, it will be a cubic polynomial.A polynomial is said to be linear, quadratic, cubic or biquadratic according to the degree of the polynomial.

The correct option is (b) 0

Explanation: α and β are the zeros of X^2 + (k^2 – 1)x – 20

So, α + β = -(k^2 – 1) and αβ = -20

Also, α – β = 9

Now, α^2 – β^2 – αβ = -11

(α + β)(α – β) – αβ = 29

-(k^2 – 1)9 + 20 = 29

-(k^2 – 1)9 = 9

-(k^2 – 1) = 1

(k^2 – 1) = -1

k^2 = -1 + 1

k = 0

The correct choice is (d) 3

The EXPLANATION is: Since, one ZERO of the POLYNOMIAL is the additive inverse of the other.

Hence, the sum of roots will be zero.

The polynomial is x^2+(k-3)x-16=0

Sum of ZEROS or α+β=\(\frac {-coefficient \, of \, x}{coefficient \, of \, x^2} = \frac {k-3}{1}\)=0

k-3=0

k=3

The CORRECT answer is (b) -10x^3-15x^2+10x+20

The explanation is: We KNOW that,

F(x)=q(x)×G(x)+r(x)

Where, f(x) is the dividend, q(x) is the quotient, g(x) is the divisor and r(x) is the remainder.

f(x)=5x^2×(2x+3)+10x+20

f(x)=10x^3+15x^2+10x+20

The correct CHOICE is (a) x^2+2x+5

To explain: An expression in the form of (x)=a0+a1x+a2x^2+…+anx^n, where an≠0, is called a polynomial where a1, a2 … an are real NUMBERS and each power of x is a non-negative integer.

In case of √x+2x+4 , the power of √x is not an integer. Similarly for x^\(\frac {2}{3}\)+10x, \(\frac {2}{3}\) is a fraction.

Now, 5X+\(\frac {5}{x}\) in this case the power of x is a negative integer. Hence it is not a polynomial.

Right option is (d) \(\frac {1}{4}\)

For explanation: \(\frac {1}{\alpha } + \frac {1}{\BETA } = \frac {\alpha +\beta }{\alpha \beta }\)

α+β=\(\frac {-20}{10}\)=-2

αβ=\(\frac {-80}{10}\)=-8

∴ \(\frac {\alpha +\beta }{\alpha \beta } = \frac {-2}{-8} = \frac {1}{4}\)

The correct answer is (b) √3

To EXPLAIN: α and β are the zeros of x^2 – (5 + 7k)x + 35K

So, α + β = (5 + 7k) and αβ = 35k

Also, α^2 + β^2 = 172

(α + β)^2 – 2αβ = 172

Substituting VALUES we get,

(5 + 7k)^2 – 2(35k) = 172

25 + 49k^2 + 70K – 70k = 172

49k^2 – 147 = 0

Solving we get, k = √3