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Correct answer is (c) (X + 2y + z) (x^2 + 4y^2 + z^2 – 2xy – 2YZ – zx)

Explanation: We know that a^3 + b^3 + c^3 – 3ABC = (a+b-c)(a^2 + b^2 + c^2 – ab – bc – ca)

In x^3 + 8y^3 + y^3 – 6xyz, a=x, b=2y and c=z

By using the above EQUATION, we get x^3 + 8y^3 + z^3 – 4xyz = (x+2y+z) (x^2 + (2y)^2 + z^2 – x(2y) – (2y)(z) – zx)

= (x + 2y + z) (x^2 + 4y^2 + z^2 – 2xy – 2yz – zx).

Right option is (d) p^2 + 9q^2 + 4z^2 + 6PQ – 12qz – 4zp

Explanation: We know that (a+b+C)^2 = a^2 + b^2 + c^2 + 2AB + 2BC + 2ca

(p+3Q-2z)^2can also be written as (p+3q+(-2z))^2.

Here, a = p, b = 3q and c = -2z

Therefore, using that formula we get (p+3q+(-2z))^2

= p^2 + (3q)^2 + (-2z)^2 + 2(p)(3q) + 2(3q)(-2z) + 2(-2z)(p)

= p^2 + 9q^2 + 4z^2 + 6pq – 12qz – 4zp.

The CORRECT option is (c) 857375

Easy explanation: We know that (x-y)^3 = x^3– y^3 – 3 xy (x-y).

95^3can ALSO be written as (100-5)^3

Therefore, 95^3= (100-5)^3 = (100)^3 – (5)^3 – 3(100)(5)(100-5)

= 1000000 – 125 – 1500(95)

= 1000000 – 125 – 142500

= 857375.

Right option is (B) 884

To ELABORATE: 26*34 can ALSO be written as (30-4)*(30+4)

We KNOW that (a-b)*(a+b) = a^2 -b^2

Similarly, 26*34 = (30-4)*(30+4)

= 30^2 – 4^2

= 900 – 16

= 884.

Correct choice is (a) (x+9)(x-3)

To elaborate: To factorisex^2+6x-27, we have to FIND TWO numbers ‘a’ and ‘b’ such that a+b=6 and a*b=27.

For that we have to find factors of -27, which are ±1, ±3, ±9.

Now we have to ARRANGE two numbers from these numbers such that a+b=6 and a*b=27.

By considering this, we get two numbers +9 and -3

9 + (-3) = 6 and 9*-3 = -27

Now after manipulating terms, we getx^2+9x- 3x-27.

x^2+9x- 3x-27 = x(x+9)-3(x+9)

= (x+9)(x-3).

The CORRECT option is (d) -120

The explanation is: According to factor theorem, x-a is a factor of P(x) if p(a) = 0.

Here, it is GIVEN that x-3 is a factor of 5x^3-2x^2+x+k.

Therefore, p(3) must be EQUAL to zero.

p(3) = 5(3)^3-2(3)^2+3+k = 0

Therefore, 5(27) – 2(9) + 3 + k = 0

135 – 18 + 3 + k=0

120 + k = 0

Therefore,k= -120

The correct choice is (a) True

To explain: p(x) is SAID to be multiplier of x-a if p(a) = 0.

Therefore, CHECKING whether p(-4) is ZERO of p(x) or not.

p(-4) = (-4)^2 + 8(-4) + 16

= 16 – 32 +16

= 32 – 32

= 0

p(-4) = 0, so we can SAY that x^2 + 8x + 16 is a multiplier of x+4.

Or in other words, we can say that there we get “0” as a remainder when we divide x^2 + 8x + 16 by x+4.

Correct choice is (b) False

Explanation: ACCORDING to factor theorem, x-a is a factor of P(x) if p(a) = 0.

Therefore x-1 is a factor of 4x^2-9x-6 is a factor if p(1)=0.

p(1) = 4(1)^2-9(1)-6 = 4 – 9 – 6

= -11 ≠ 0

Therefore, we can say that x-1 is not a factor of 4x^2-9x-6.

Correct choice is (b) 52

The best explanation: According to remainder theorem, if p(X) be any polynomial of degree greater than or equal to one and LET “a” be any real number. If p(x) is DIVIDED by the linear polynomial x-a, then the remainder is p(a).

So to know the value of remainder, we have to find the value of p(2).

p(2) = 6(2)^3 + (2)^2 – 2(2) + 4

= 6(8) + 4 – 4 + 4

= 48 + 4

= 52.

Correct option is (a) -3

Easy explanation: If p(X) is any POLYNOMIAL, then for “a” to be a zero of the polynomial p(x), p(a) = 0.

Therefore, equating p(x) with 0, we GET x^2+6x+9=0

(x+3)^2 = 0

x+3 = 0

x = -3

Hence, -3 is the zero of the polynomial x^2+6x+9 because p(-3) = 0.

Right option is (c) 22

For explanation: If p(x) is any polynomial, then p(a) MEANS substituting x by a in a polynomial.

Therefore, p(2) means, putting 2 in PLACE of x in 4x^2+7x-8.

Hence, p(2) = 4(2)^2+7(2)-8 = 4(4) + 14 – 8

= 16 + 14 – 8

= 22.

The correct answer is (c) 3

Easiest explanation: We can GET number of TERMS by separating them by “+” or “–” SIGN.

In the GIVEN polynomial 5x^2+2X-2, if we separate expressions by “+” or “–” sign, we get three separate terms as 5x^2, 2x and 2.

Hence we can say that there are three terms in the polynomial 5x^2+2x-2.

The correct answer is (d) 5x^2+2x-2

The best I can explain: Polynomials only having THREE terms are CALLED Trinomials.

Hence to find trinomial POLYNOMIAL, we have to find polynomial which has three terms. We can see that 5x^2+2x-2 has three terms namely 5x^2, 2x and 2. Hence it is trinomial.

Similarly, polynomials only having ONE TERM are called Monomials and polynomials only having two terms are called Binomials.

The correct ANSWER is (B) x^2 – 2xy+y^2

Easy EXPLANATION: We can write (x-y)^2 as (x-y) * (x-y) = x^2-xy-yx + y^2

= x^2-2xy + y^2.

The correct ANSWER is (d) 3

The EXPLANATION is: A polynomial of degree 2 is called quadratic polynomial.

Quadratic polynomials are of the form ax^2+BX+c and it can contain at most three terms namely ax^2, bx and c. Thus, we can say that a quadratic polynomial can have at most three terms.

Similarly, a polynomial of degree 1 is called LINEAR polynomial and a polynomial of degree 3 is called CUBIC polynomial.

The correct answer is (a) Not DEFINED

To EXPLAIN: Degree of the ZERO POLYNOMIAL is not defined.

Zero polynomial is denoted by 0, and degree for that is not defined.

Right choice is (C) 0

Explanation: DEGREE of a non-zero constant polynomial is zero.

We can see that given polynomial 7 CONTAIN only ONE term and that is constant. 7 can also be written as 7x^0.

Hence degree of 7 is zero.

Correct OPTION is (B) False

The BEST I can explain: For an expression to be a polynomial, exponent of variable has to be whole NUMBER.

\(\frac{1}{\sqrt[2]{x}}\) can be written as x^-1/2. We can see that exponent of x is -1/2 which is not whole number (W = {0, 1, 2, 3…}). Hence, \(\frac{1}{\sqrt[2]{x}}\) is not a polynomial.

Correct choice is (a) 7

The best EXPLANATION: Degree of a polynomial is the HIGHEST power of variable in a polynomial.

The TERM with the highest power of x is 4x^7 and EXPONENT of x in that term is 7, so the degree of polynomial of 4x^7+ 9x^5+5x^2+11 is 7.

Correct answer is (d) 0

The EXPLANATION: Coefficient is the NUMBER which is multiplied with respective variable.

In the given POLYNOMIAL 6x^4 + 3x^2 + 8x + 5, there is not an expression containing x^3. So we can write 6x^4+ 3x^2+8x+5 as 6x^4 + 0x^3 + 3x^2 + 8x + 5. We can SEE that 0 is multiplied with expression x^3, so coefficient of x^3 is 0.