# IDENTITIES

1. If x(x-4)(2x+3) is an identity of Px^3 + Qx^2 +Rx + S for all values of x, find the values of P, Q, R and S.

2. If 3x^2 - 7x + 6 is an identity of Ax(x-1) + Bx(x-2) + C(x-1)(x-2), what are the values of A, b and C?

3. If Px^2 + Q(x+2) - R(x-3) is an identity of (x+3)(3x-2), find the values of P, Q and R.

### 2 個解答

• 最愛解答

1. If x(x-4)(2x+3) is an identity of Px^3 + Qx^2 +Rx + S for all values of x, find the values of P, Q, R and S.

x(x-4)(2x+3) ≡ Px³ + Qx² +Rx + S

L.H.S. = (x² - 4x)(2x + 3)

..........= 2x³ + 3x² - 8x² - 12x

R.H.S. = Px³ + Qx² +Rx + S

2x³ + 3x² - 8x² - 12x ≡ Px³ + Qx² +Rx + S

By comparing the coefficient,

P = 2 , Q = 3, R = -8, S = 0

2. If 3x^2 - 7x + 6 is an identity of Ax(x-1) + Bx(x-2) + C(x-1)(x-2), what are the values of A, b and C?

3x² - 7x + 6 ≡ Ax(x-1) + Bx(x-2) + C(x-1)(x-2)

L.H.S. = 3x² - 7x + 6

R.H.S. = Ax(x-1) + Bx(x-2) + C(x-1)(x-2)

...........= Ax² - Ax + Bx² - 2Bx + C(x² - 3x + 2)

...........= Ax² - Ax + Bx² - 2Bx + Cx² - 3Cx + 2C

...........= Ax² + Bx² + Cx² - Ax - 2Bx - 3Cx + 2C

...........= ( A+B+C)x² - (A + 2B + 3C)x + 2C

3x² - 7x + 6 ≡ ( A+B+C)x² - (A + 2B + 3C)x + 2C

By comparing the constant term ,

2C = 6

C = 3 //

By comparing the coefficient of x²

A + B + 3 = 3

A + B = 0......................(1)

By comparing the coefficient of x

A + 2B + 9 = 7

A + 2B = -2.....................(2)

(2) - (1):

A + 2B - A - B = -2 - 0

B = -2 //

Substitute B = -2 into (1),

A - 2 = 0

A = 2 //

A = 2, B = -2, C = 3

3. If Px^2 + Q(x+2) - R(x-3) is an identity of (x+3)(3x-2), find the values of P, Q and R

Px² + Q(x+2) - R(x-3) ≡ (x+3)(3x-2)

L.H.S. = Px² + Qx + 2Q - Rx + 3R

..........= Px² + (Q-R)x + 2Q + 3R

R.H.S. = (x+3)(3x-2)

...........= 3x² - 2x + 9x - 6

...........= 3x² + 7x - 6

Px² + (Q-R)x + 2Q + 3R ≡ 3x² + 7x - 6

By comparing the coefficient of x²

P = 3 //

By comparing the coefficient of x

Q - R = 7.............(1)

By comparing the constant term,

2Q + 3R = -6........(2)

From (1),

Q = R + 7.............(3)

Substitute (3) into (2)

2( R+7 ) + 3R = -6

2R + 14 + 3R = -6

5R = -20

R = -4 //

Substitute R = -4 into (3)

Q = -4 + 7

Q = 3 //

P = 3, Q = 3, R = -4

2007-02-11 13:02:57 補充：

小提示: 有時 D identity 的題目, 計完之後, 唔可以直接找出答案, 所以我地要將整條式 expand 左佢, 再 factorize D like terms, 然後仲要用二元一次方程式解 ~( simultaneous equations in two unknowns )第二, 三題就係一個gd. example, 由於需要寫好多 steps , 所以需要好小心, 答完最好都係驗算多幾次, 咁就萬無一失喇~

資料來源： me~
• 1.If x(x-4)(2x+3) is an identity of Px^3 + Qx^2 +Rx + S for all values of x, find the values of P, Q, R and S.

LHS=x(x-4)(2x+3)

------=(x^2-4x)(2x+3)

------=2x^3-8x^2+3x^2-12x

------=2x^3-5x^2-12x

RHS=Px^3+Qx^2+Rx+S

Compare the coefficient of x^3 , x^2 , x^1 and the constant P,Q,R,S =

2 , -5 , -12 & 0 respectively.

2.If 3x^2 - 7x + 6 is an identity of Ax(x-1) + Bx(x-2) + C(x-1)(x-2), what are the values of A, b and C?

LHS=3x^2 - 7x + 6

RHS=Ax(x-1) + Bx(x-2) + C(x-1)(x-2)

------=Ax^2-Ax + Bx^2-2Bx + (Cx-C)(x-2)

------=(A+B)x^2 - (A+2B)x + Cx^2 - Cx - 2Cx + 2C

------=(A+B+C)x^2 - (A+2B+3C)x + 2C

Compare the coefficient of x^2 , x^1 and the constant

2C = 6

C=3

(A+B+C)x^2 = 3x^2

(A+B+3)x^2 = 3x^2

(A+B+3) = 3

A+B=0------(1)

(A+2B+3C)x =7x

(A+2B+9)=7

A + 2B = -2-----(2)

(2) - (1)

(A + 2B) - (A+B) = -2 - 0

B = -2

Sub B = -2 into (1)

A - 2 =0

A = 2

∴A = 2 , B = -2 , C = 3

3. If Px^2 + Q(x+2) - R(x-3) is an identity of (x+3)(3x-2), find the values of P, Q and R.

LHS= Px^2 + Q(x+2) - R(x-3)

------= Px^2 + Qx + 2Q – Rx + 3R

------= Px^2 + x(Q-R) + 2Q + 3R

RHS= (x+3)(3x-2)

------=3x^2 + 9x – 2x -6

----- =3x^2 + 7x -6

Compare the coefficient of x^2 , x^1 and the constant

Px^2 = 3x^2

P=3

x(Q-R) = 7x

Q – R = 7

Q = 7 +R ------(1)

2Q + 3R = -6 -----(2)

Sub (1) into (2)

2 (7+R) + 3R = -6

14 + 5R = -6

R = -4

Sub R = 4 into (1)

Q = 7 - 4

Q =3

∴P=3 , R = -4 , Q =3