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Polynomials, , Quick Revision, , A polynomial in one variable x, is an algebraic, expression of the form, / n n=l n=? 2, p(x) = a,x FO RO Fy gh Hoe F Ayd, +a,x+ay, , where 7 is a positive integer and constants, Mo, 4, @y,...,@, are known as coefficients of, , polynomial., Degree of a Polynomial, , The highest power (exponent) of x ina, polynomial f(x), is called the degree of the, polynomial f(x)., , eg. g(y)= 3y? - y +7is a polynomial in, , variable y of degree 2., , Types of Polynomials, (i) Linear Polynomial A polynomial of degree, one, is called linear polynomial., eg. f(x)=3x4+5, (ii) Quadratic Polynomial A polynomial of, degree two, is called quadratic polynomial., , 7, , eg. f(x) = 5x? + 3x, , , , (iii) Cubic Polynomial A polynomial of degree, three, is called cubic polynomial., eg. f(x) =9x" + 5x7, , {iv) Biquadratic Polynomial A polynomial, of degree four, is called biquadratic, polynomial., eg. f(x)=at+ox?, , , , , x + 5x43, , Value of a Polynomial at Given Point, , If p(x) is a polynomial and a is a real value, then the, value obtained by putting x =a in p(x) , is called, the value of p(x) al x =a and itis denoted by f(a)., Zeroes of a Polynomial, , Areal number £ is said to be a zero of a polynomial, f(x), if f(k) = 0., , Geometrical Meaning of the Zeroes of a, Polynomial, , ‘The geometrical meaning of the zeroes of a, polynomial means that the curve intersect the, X-axis, the intersection point is said to be zeroes of, the curve., , e.g. In the figure we see that graph intersect the, X-axis al points A, B, Cand D. So, it has four zeroes., , , , Relationship between Zeroes and, Coefficients of a Polynomial, The zeroes of a polynomial are related to its, coefficients., (i) For a Linear Polynomial The zero of the, linear polynomial ax + 6 is, , 6 Constant term, , a Coefficient of x

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18, , (i) For a Quadratic Polynomial Let u and $B be, , the zeroes of quadratic polynomial, p(x) = ax" + bx +c,a £0, then, , «. Sum of zeroes, & + B, , __ Coefficientof x 6, ~~ Coefficient of x2 a, and product of zeroes, oB, Constant term __¢, , , , ~ Coellicient ofx? a, (iii) For a Cubic Polynomial Let «, B and 7 be, , the zeroes of the cubic polynomial, ax? + bx” + ex + d,a 40), then, , Coefficient of x?, , Cena Coefficient of «*, _ ob, _ a, dey e= Coefficient of _ _¢, Coefficient of x” @, , Constant term, Coefficient of x*, d, , a, , and apy =, Useful Algebraic Identities, (a+b) =a’ +b? + 2ab, (ii) (a + 6)? — (a — O° =4ab and, (a+b)? +(a— 6)? = %a? +8"), , CBSE New Pattern ~ Mathematics X (Term 1D), , (iii) (a? — b?) = (a+ b) (a — 6), , fiv) (a+ b+0)? =a? +8? +06? +2 (ab + be + ac), , (v) (a? + 6°) =(a +b) (a? +B? F ab), =(a+6)[(a + 4)° F 32d], , (vi) (a + 8)° = a* +8° + Baba +3), =a" +6 + 3a"b + 3ab?, , — b° — 3ab(a — 6), , =a’ —b — 3a°b + 3ab”, , , , (vii) (a - 6)°, , (viii)a? + 6° +c? — 3abe, =(a+b4+c}(a? +b? +07 - ab - be - ca), , Formation of Quadratic and, Cubic Polynomials, , (i) Ito and B are the zeroes of a quadratic, polynomial, then quadratic polynomial will be, , k [x?~ (sum of zeroes) x + product of zeroes], ie. k |x? — (a 4 B)x + oP], where & is some, constant., (ii) Ifa, B and y are the zeroes of cubic, polynomial, then cubic polynomial will be, lx? — (sum of zeroes) x”, + (sum of the product of zeroes, taking two at a time)x, — product of zeroes], , ie. klx?— (a+ B+y)x° + (aB + Byt yo) x-oBy]

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Objective Questions, , Multiple Choice Questions, , 1. Which of the following is not a, , polynomial, (a)3x45 (b) 3y3—4y? +2y, (c)x-3 (4), , X+2, , 2. If 2isa zero of polynomial, I(x) = ax” — 3a —1)x - 1, then the value, , of ais, , (a)0 (b}2, 5 1, (chs (de, , 3. Lf one of the zeroes of the quadratic, polynomial (k — Dx? + kx +1is - 3, then, , the value of kis INCERT Exemplarj, (a) (b), 3 3, 2 oe, (oe (a), ae U3, 4. If one of the zeroes of the quadratic, , polynomial x? + 3x + kis 2, then the, , value of kis [CBSE 2020], (a) 10 (b)-10, (c)-7 (dj-2, , 5. If 2 and 3 are zeroes of polynomial, 3x” — ke + 2m, then the values of k and, , mare, , (alm=Sandk='5 iota- 2 andk=9, , 5, , (clm=9ancK= 2 (d)m=15 andk =9, , 6. The value of f, for which (—4) is a zero, , of the polynomial x? — 2x — (7 p + 3) is, , , , ( (b}2, (c)4 (dj-2, , 7. If the graph of a polynomial intersects, the X-axis at only one point, it can be a, , 10., , 11., , 12., , 13., , {a} linear, (c) cubic, , (b) quadratic, (d) None of these, , » Which of the following is not the graph, , of a quadratic polynomial?, , , , , , V, , , , , , . The graph of a quadratic polynomial, , 1S Seveeemsven,, , la)st ate, (c) hyperbola, , INCERT Exemplar], , (b) parabola, (d) None of these, , , , If one zero of polynomial x’ — 4x + Lis, 2 + V3, then other zero will be ........, (a)-24+V3 (b)- V3 -2, , (c)2—V3 (d)V¥3+, , ihe zeroes of the quadratic polynomial, (x? +5x + 6) are, , , , (a)-2and-3 (b)3and4, (c)3and 2 (d)2 and1, Zeroes of p{z)= * _97are.., , ANG ceneceennes, , lal 3v3 (b)+3 :, (c)+9 (d)+ V3 and—V3, , The zeroes of the quadratic polynomial, f (x) = abx? UP me a, (a) and © tb) and @, , ac b c b, , DY al b -c, (land (d) ands

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20, , 14. The number of polynomials having, , 15., , 16., , 17., , 18., , 19., , 20., , 21, , 22., , zeroes as—2and5is [NCERT Exemplar], (a) (b)2, , (c)3 (d) more than 3, , J and 2 are the zeroes the polynomial, x? 342, , (a) lrue (b) False, , (c) Can't say (d) Partially true/false, Every real number is the zeroes of zero, polynomial, , (a) True (b} False, , (c) Can't say (d) Partially true/False, , pix) =x land g(x) =x? — 2x +1, plx), is a factor of g(x), , (e) True (b) False, , (c) Can't Say (d) Partially True/False, , The value of k for which 3 is a zero of, polynomial 2x? +4 + his., (a)21 (b}20, , (c)21 (d)18, , If zeroes « and B of a polynomial, , x? —7x +kare such that 0 —B =] then, the value of & is, , (a) 21 (b)12, , (c)9 (d8, , Sum of zeroes of Quadratic polynomial, Coefficient of x, , Coefficient of x”, (a) True (b) False, (c) Can‘t say (d) Partially true/false, , Ifa and B are zeroes of the quadratic, polynomial x” — 6x +a the value of ‘a,, if 3a. + 2B = 20 is - 12, , (a) True (b) False, , (c) Can’t say (d} Partially True/False, If one of the zeroes of a quadratic, polynomial of the form x? +ax + bis, , the negative of the other, then it, , (a) has no linear term and the constant term is, negative, , CBSE New Pattern ~ Mathematics X (Term 1D), , (b) has no linear term and the constant term is, positive, , (c) can have a lineer term but the constant, term is negative, , (c) can have a linear term but the constant, term is positive, , 23. If pand g are zeroes of 3x” +2xv-9,, then value of p- 4 is, , 2, {a)-3 (b), (c)e70* (d) Nane of these, , 24. The polynomial whose zeroes are, (V2 +1) and (V2 —1) is, , (a)x? +2V2x41 (b) x? 22x41, (co) x? +2V2x-1 (d) x? -22x-1, , 25. Ifa and Pare the zeroes of the, polynomial Qy? +7y + 5, then the value, ofa +B+oaPis, (a)-1 (b)0, {c}1 (d)2, , 26. Ifo and B are the zeroes of the, quadratic polynomial, f(x) = 8x? —5x —2, then a +B? is, , equal to ., ir 7, (yy a, 27. Ifa and fare the zeroes of, 4x? + 3x +7, then the value of + ; is, a, (v) 2, ols (ae, , 28. If the zeroes of the quadratic polynomial, x? +(a+I)x + bare 2 and — 3, then, INCERT Exemplar], (b)a=5,b=-1, (d)a=0,b =-6

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CBSE New Pattern ~ Mathematics X (Term 1), , 29., , 30., , 31., , 32., , 33., , 34., , 35., , If the sum and difference of zeroes of, quadratic polynomial are — 3 and — 10,, respectively. Then, the difference of the, squares of zeroes is, , (a)20 (b} 30, (c)15 (d} 25, If sum and product of zeroes of, , quadratic polynomial are, respectively, 8 and 12, then their zeroes are, , (a)2 and6 (b}3 and 4, (c)2 and 8 (dj2and5, If m and vare the zeroes of the, , polynomial 3x? +1 1x — 4, then find the, , mon, value of — +—, n, , , , m, , 145 145, @ bh, ( lag (b) G, , 145 —145, (c)—— d}——, c) 7 (d} 15, If one zero of the polynomial, , 3x? — 8x + 2k +is seven times the, other, then the value of £ is, , 2 2, , (ale (b}, , (o)% ae, z, , If sum of the squares of zeroes of the, quadratic polynomial, , f(x)= x? —4x+ kis 20, then the value, of kis, , (a)-2 (b)—3, (c)-4 (d}2, Ifa and B are zeroes of the polynomial, , x? — p(x +1) +e such that, (a +1)(8 +1) =0, then the value of ¢ is, , (a)-2 (b}2, (c)-1 (d}1, The value of k such that the polynomial, , x? —(k +6)» + 2(2k —1) has sum of its, , zeroes equal to half of their product is, [CBSE 2019}, , (a)-4 (b)4 (c}-7 (d)7, , 36., , 37., , 38., , 39., , 40., , 41., , 42., , 43., , 21, , . The sum and the product of zeroes of, the polynomial f(x) = 4”? — 27x + 3k?, are equal the value of k is, , (alk=3 (b)k=-3, , (c)k=43 (d)k=2, , Ifa and B are the zeroes of the, quadratic polynomial, , f(x) =x? —4x +3, then the value of, a'B* +0°B" is, , (a) 104 (b) 108, , (c) N12 (d)5, , The quadratic polynomial, the sum of, whose zeroes is — 5 and their product is, 6, is ICBSE 2020], (a)x +5x+6 (b)x¢ -5x+6, , (c)x’ -5x-6 (d)—x’ +5x+6, , The quadratic polynomial whose zeroes, are 27 and -5v7 is, (a)x’ -3V7x-70, (ce)? + 3V7x-70, , (b)x? + 3V7x+ 70, (d)x’ - 3V7x+70, The quadratic polynomial, whose, zeroes are 3 + V2 and 3 ~ VQ, is, (b)x? - 8x +7, (d)x° -8x+12, , (a)? -3x+5, (oc) -7x+6, Ifa and B are the zeroes of the, quadratic polynomial f(x) =x? +x - 2,, then the polynomial whose zeroes are, 20 + land 2B +1is, , lal +9, (ce) -8, , (b)x? 4, , (d)x’ +4, , Ifo and B are zeroes of a quadratic, polynomial x? —5, then the quadratic, polynomial whose zeroes are 1+ and, 1+Bis, , (a)? +2x+24, (c) xf -2x+24, , (b)x° -2x-24, , (d) None of these, The number of value of & for which the, quadratic polynomial ke? +4 + khas, , equal zeroes is, , (a) 4 (b)1 (c)2 (d)3