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Question Number 216820 by Ismoiljon_008 last updated on 22/Feb/25

$$$$$$f\left(x\right)={ax}^4+{bx}^3+{cx}^2+dx+e$$$$f\left(1\right)=2$$$$f\left(2\right)=3$$$$f\left(3\right)=4$$$$f\left(4\right)=5$$$$f\left(0\right)=25$$$$\mathcal{T}henf\left(5\right)=?$$$$\mathcal{H}elpme,please$$$$$$

Answered by BaliramKumar last updated on 22/Feb/25

Commented by Ismoiljon_008 last updated on 22/Feb/25

$$thankyouverymuch$$

Commented by ArshadS last updated on 22/Feb/25

$$\mathrm{Some}\mathrm{explanation}\mathrm{please}!$$

Answered by mr W last updated on 22/Feb/25

$$f\left(1\right)=2=1+1$$$$f\left(2\right)=3=2+1$$$$f\left(3\right)=4=3+1$$$$f\left(4\right)=5=4+1$$$$wecansee$$$$f\left(x\right)=x+1+g\left(x\right)with$$$$g\left(x\right)=polynomialof{4}^{th}degreeand$$$$g\left(1\right)=g\left(2\right)=g\left(3\right)=g\left(4\right)=0$$$$\Rightarrow g\left(x\right)=A(x-1)(x-2)(x-3)(x-4)$$$$f\left(x\right)=x+1+A(x-1)(x-2)(x-3)(x-4)$$$$f\left(0\right)=1+A(-1)(-2)(-3)(-3)=1+24A=25$$$$\Rightarrow A=1$$$$\Rightarrow f\left(x\right)=x+1+(x-1)(x-2)(x-3)(x-4)$$$$f\left(5\right)=5+1+4\times 3\times 2\times 1=30✓$$

Commented by ArshadS last updated on 23/Feb/25

$$thankssir!$$$$f\left(1\right),f\left(2\right),f\left(3\right),f\left(4\right)allagreethepattern:f\left(x\right)=x+1$$$$butf\left(0\right)=24disagrees.Thenwhyingeneralthe$$$$patternf\left(x\right)=x+1ismade?$$

Commented by mr W last updated on 23/Feb/25

$$wecannotsayf\left(x\right)=x+1.butwe$$$$cansay$$$$f\left(x\right)=x+1+A(x-1)(x-2)(x-3)(x-4)$$

Commented by ArshadS last updated on 23/Feb/25

$$\mathcal{T}hankstoteachmesir!$$

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