For algebra exercises in PDF format, consider:
1.1 Suma y resta
a) (P(x)=3x^4-5x^3+2x^2-7)
b) (Q(x)=-2x^4+4x^3+x^2+5x-1)
Ejercicio: Calcula (P(x)+Q(x)) y (P(x)-Q(x)).
Solución:
[ \beginaligned P(x)+Q(x) &= (3x^4-2x^4)+(-5x^3+4x^3)+(2x^2+x^2)+(0+5x)+(-7-1)\ &= x^4-x^3+3x^2+5x-8. \endaligned ]
[ \beginaligned P(x)-Q(x) &= (3x^4+2x^4)+(-5x^3-4x^3)+(2x^2-x^2)+(0-5x)+(-7+1)\ &= 5x^4-9x^3+x^2-5x-6. \endaligned ]
1.2 Producto de polinomios
(A(x)=x^2+3x-4)
(B(x)=2x-5)
Ejercicio: Calcula (A(x)\cdot B(x)).
Solución:
[ \beginaligned A(x)B(x) &= (x^2+3x-4)(2x-5)\ &= x^2(2x-5)+3x(2x-5)-4(2x-5)\ &= 2x^3-5x^2+6x^2-15x-8x+20\ &= 2x^3+x^2-23x+20. \endaligned ]
5.1 Suma y multiplicación de matrices
[ M= \beginpmatrix 1 & 2\ 3 & 4 \endpmatrix, \qquad N= \beginpmatrix 0 & -1\ 5 & 2 \endpmatrix. ]
5.2 Determinante de una matriz 3×3
[ A= \beginpmatrix 2 & -1 & 3\ 0 & 4 & 5\ -2 & 1 & 0 \endpmatrix ]
[ \det(A)=2\beginvmatrix4&5\1&0\endvmatrix -(-1)\beginvmatrix0&5\-2&0\endvmatrix +3\beginvmatrix0&4\-2&1\endvmatrix =2(4\cdot0-5\cdot1)+1(0\cdot0-5\cdot(-2))+3(0\cdot1-4\cdot(-2)) ]
[ =2(-5)+1(10)+3(8)= -10+10+24=24. ]
[ \boxed\det(A)=24. ]